1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: decodegb.lib 15103 2012-07-11 10:00:13Z motsak $"; |
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3 | category="Coding theory"; |
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4 | info=" |
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5 | LIBRARY: decodegb.lib Decoding and min distance of linear codes with GB |
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6 | AUTHOR: Stanislav Bulygin, bulygin@mathematik.uni-kl.de |
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7 | |
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8 | OVERVIEW: |
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9 | In this library we generate several systems used for decoding cyclic codes and |
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10 | finding their minimum distance. Namely, we work with the Cooper's philosophy |
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11 | and generalized Newton identities. The origindeal method of quadratic equations |
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12 | is worked out here as well. We also (for comparison) enable to work with the |
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13 | system of Fitzgerald-Lax. We provide some auxiliary functions for further |
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14 | manipulations and decoding. For an overview of the methods mentioned above @ref{Decoding codes with Groebner bases}. |
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15 | For the vanishing ideal computation the algorithm of Farr and Gao is |
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16 | implemented. |
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17 | |
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18 | PROCEDURES: |
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19 | sysCRHT(..); generates the CRHT-ideal as in Cooper's philosophy |
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20 | sysCRHTMindist(..); CRHT-ideal to find the minimum distance in the binary case |
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21 | sysNewton(..); generates the ideal with the generalized Newton identities |
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22 | sysBin(..); generates Bin system using Waring function |
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23 | encode(x,g); encodes given message x with the given generator matrix g |
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24 | syndrome(h,c); computes a syndrome w.r.t. the given check matrix |
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25 | sysQE(..); generates the system of quadratic equations for decoding |
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26 | errorInsert(..); inserts errors in a word |
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27 | errorRand(y,num,e); inserts random errors in a word |
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28 | randomCheck(m,n,e); generates a random check matrix |
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29 | genMDSMat(n,a); generates an MDS (actually an RS) matrix |
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30 | mindist(check); computes the minimum distance of a code |
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31 | decode(rec); decoding of a word using the system of quadratic equations |
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32 | decodeRandom(..); a procedure for manipulation with random codes |
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33 | decodeCode(..); a procedure for manipulation with the given code |
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34 | vanishId(points); computes the vanishing ideal for the given set of points |
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35 | sysFL(..); generates the Fitzgerald-Lax system |
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36 | decodeRandomFL(..); manipulation with random codes via Fitzgerald-Lax |
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37 | |
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38 | |
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39 | KEYWORDS: Cyclic code; Linear code; Decoding; |
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40 | Minimum distance; Groebner bases, decodeGB |
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41 | "; |
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42 | |
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43 | LIB "linalg.lib"; |
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44 | LIB "brnoeth.lib"; |
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45 | |
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46 | /////////////////////////////////////////////////////////////////////////////// |
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47 | // creates a list result, where result[i]=i, 1<=i<=n |
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48 | static proc lis (int n) |
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49 | { |
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50 | list result; |
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51 | if (n<=0) {print("ERRORlis");} |
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52 | for (int i=1; i<=n; i++) |
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53 | { |
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54 | result=result+list(i); |
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55 | } |
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56 | return(result); |
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57 | } |
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58 | |
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59 | /////////////////////////////////////////////////////////////////////////////// |
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60 | // creates a list of all combinations without repititions of m objects out of n |
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61 | static proc combinations (int m, int n) |
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62 | { |
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63 | list result; |
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64 | if (m>n) {print("ERRORcombinations");} |
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65 | if (m==n) {result[size(result)+1]=lis(m);return(result);} |
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66 | if (m==0) {result[size(result)+1]=list();return(result);} |
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67 | list temp=combinations(m-1,n-1); |
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68 | for (int i=1; i<=size(temp); i++) |
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69 | { |
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70 | temp[i]=temp[i]+list(n); |
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71 | } |
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72 | result=combinations(m,n-1)+temp; |
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73 | return(result); |
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74 | } |
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75 | |
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76 | |
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77 | /////////////////////////////////////////////////////////////////////////////// |
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78 | // the polynomial for Sala's restrictions |
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79 | static proc p_poly(int n, int a, int b) |
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80 | { |
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81 | poly f; |
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82 | for (int i=0; i<=n-1; i++) |
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83 | { |
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84 | f=f+Z(a)^i*Z(b)^(n-1-i); |
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85 | } |
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86 | return(f); |
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87 | } |
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88 | |
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89 | /////////////////////////////////////////////////////////////////////////////// |
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90 | |
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91 | proc sysCRHT (int n, list defset, int e, int q, int m, list #) |
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92 | "USAGE: sysCRHT(n,defset,e,q,m,[k]); n,e,q,m,k are int, defset list of int's |
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93 | @format |
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94 | - n length of the cyclic code, |
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95 | - defset is a list representing the defining set, |
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96 | - e the error-correcting capacity, |
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97 | - q field size |
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98 | - m degree extension of the splitting field, |
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99 | - if k>0 additional equations representing the fact that every two |
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100 | error positions are either different or at least one of them is zero |
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101 | @end format |
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102 | RETURN: the ring to work with the CRHT-ideal (with Sala's additions), |
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103 | containig an ideal with name 'crht' |
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104 | THEORY: Based on 'defset' of the given cyclic code, the procedure constructs |
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105 | the corresponding Cooper-Reed-Heleseth-Truong ideal 'crht'. With its |
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106 | help one can solve the decoding problem. For basics of the method @ref{Cooper philosophy}. |
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107 | SEE ALSO: sysNewton, sysBin |
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108 | EXAMPLE: example sysCRHT; shows an example |
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109 | " |
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110 | { |
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111 | int r=size(defset); |
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112 | ring @crht=(q,a),(Y(e..1),Z(1..e),X(r..1)),lp; |
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113 | ideal crht; |
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114 | int i,j; |
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115 | poly sum; |
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116 | int k; |
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117 | if ( size(#) > 0) |
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118 | { |
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119 | k = #[1]; |
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120 | } |
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121 | |
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122 | //---------------------- add check equations -------------------------- |
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123 | for (i=1; i<=r; i++) |
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124 | { |
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125 | sum=0; |
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126 | for (j=1; j<=e; j++) |
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127 | { |
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128 | sum=sum+Y(j)*Z(j)^defset[i]; |
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129 | } |
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130 | crht[i]=sum-X(i); |
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131 | } |
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132 | |
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133 | //--------------------- field equations on syndromes ------------------ |
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134 | for (i=1; i<=r; i++) |
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135 | { |
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136 | crht=crht,X(i)^(q^m)-X(i); |
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137 | } |
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138 | |
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139 | //------ restrictions on error-locations: n-th roots of unity ---------- |
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140 | for (i=1; i<=e; i++) |
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141 | { |
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142 | crht=crht,Z(i)^(n+1)-Z(i); |
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143 | } |
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144 | |
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145 | for (i=1; i<=e; i++) |
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146 | { |
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147 | crht=crht,Y(i)^(q-1)-1; |
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148 | } |
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149 | |
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150 | //--------- add Sala's additional conditions if necessary -------------- |
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151 | if ( k > 0 ) |
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152 | |
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153 | { |
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154 | for (i=1; i<=e; i++) |
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155 | { |
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156 | for (j=i+1; j<=e; j++) |
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157 | { |
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158 | crht=crht,Z(i)*Z(j)*p_poly(n,i,j); |
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159 | } |
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160 | } |
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161 | } |
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162 | export crht; |
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163 | return(@crht); |
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164 | } |
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165 | example |
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166 | { "EXAMPLE:"; echo=2; |
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167 | // binary cyclic [15,7,5] code with defining set (1,3) |
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168 | intvec v = option(get); |
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169 | |
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170 | list defset=1,3; // defining set |
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171 | int n=15; // length |
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172 | int e=2; // error-correcting capacity |
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173 | int q=2; // basefield size |
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174 | int m=4; // degree extension of the splitting field |
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175 | int sala=1; // indicator to add additional equations |
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176 | |
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177 | def A=sysCRHT(n,defset,e,q,m); |
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178 | setring A; |
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179 | A; // shows the ring we are working in |
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180 | print(crht); // the CRHT-ideal |
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181 | option(redSB); |
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182 | ideal red_crht=std(crht); // reduced Groebner basis |
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183 | print(red_crht); |
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184 | |
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185 | //============================ |
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186 | A=sysCRHT(n,defset,e,q,m,sala); |
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187 | setring A; |
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188 | print(crht); // CRHT-ideal with additional equations from Sala |
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189 | option(redSB); |
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190 | ideal red_crht=std(crht); // reduced Groebner basis |
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191 | print(red_crht); |
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192 | red_crht[5]; // general error-locator polynomial for this code |
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193 | option(set,v); |
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194 | } |
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195 | |
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196 | /////////////////////////////////////////////////////////////////////////////// |
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197 | |
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198 | |
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199 | proc sysCRHTMindist (int n, list defset, int w) |
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200 | "USAGE: sysCRHTMindist(n,defset,w); n,w are int, defset is list of int's |
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201 | @format |
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202 | - n length of the cyclic code, |
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203 | - defset is a list representing the defining set, |
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204 | - w is a candidate for the minimum distance |
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205 | @end format |
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206 | RETURN: the ring to work with the Sala's ideal for the minimum distance |
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207 | containing the ideal with name 'crht_md' |
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208 | THEORY: Based on 'defset' of the given cyclic code, the procedure constructs |
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209 | the corresponding Cooper-Reed-Heleseth-Truong ideal 'crht_md'. With |
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210 | its help one can find minimum distance of the code in the binary |
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211 | case. For basics of the method @ref{Cooper philosophy}. |
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212 | EXAMPLE: example sysCRHTMindist; shows an example |
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213 | " |
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214 | { |
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215 | int r=size(defset); |
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216 | ring @crht_md=2,Z(1..w),lp; |
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217 | ideal crht_md; |
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218 | int i,j; |
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219 | poly sum; |
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220 | |
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221 | //------------ add check equations -------------- |
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222 | for (i=1; i<=r; i++) |
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223 | { |
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224 | sum=0; |
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225 | for (j=1; j<=w; j++) |
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226 | { |
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227 | sum=sum+Z(j)^defset[i]; |
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228 | } |
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229 | crht_md[i]=sum; |
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230 | } |
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231 | |
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232 | |
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233 | //----------- locations are n-th roots of unity ------------ |
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234 | for (i=1; i<=w; i++) |
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235 | { |
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236 | crht_md=crht_md,Z(i)^n-1; |
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237 | } |
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238 | |
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239 | //------------ adding conditions on locations being different ------------ |
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240 | for (i=1; i<=w; i++) |
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241 | { |
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242 | for (j=i+1; j<=w; j++) |
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243 | { |
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244 | crht_md=crht_md,Z(i)*Z(j)*p_poly(n,i,j); |
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245 | } |
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246 | } |
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247 | |
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248 | export crht_md; |
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249 | return(@crht_md); |
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250 | } |
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251 | example |
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252 | { |
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253 | "EXAMPLE:"; echo=2; |
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254 | intvec v = option(get); |
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255 | // binary cyclic [15,7,5] code with defining set (1,3) |
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256 | |
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257 | list defset=1,3; // defining set |
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258 | int n=15; // length |
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259 | int d=5; // candidate for the minimum distance |
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260 | |
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261 | def A=sysCRHTMindist(n,defset,d); |
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262 | setring A; |
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263 | A; // shows the ring we are working in |
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264 | print(crht_md); // the Sala's ideal for mindist |
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265 | option(redSB); |
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266 | ideal red_crht_md=std(crht_md); |
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267 | print(red_crht_md); // reduced Groebner basis |
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268 | |
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269 | option(set,v); |
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270 | } |
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271 | |
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272 | /////////////////////////////////////////////////////////////////////////////// |
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273 | // slightly modified mod function |
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274 | static proc mod_ (int n, int m) |
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275 | { |
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276 | n=n mod m; |
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277 | if (n<=0){ return(n+m);} |
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278 | return(n); |
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279 | } |
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280 | |
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281 | /////////////////////////////////////////////////////////////////////////////// |
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282 | |
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283 | proc sysNewton (int n, list defset, int t, int q, int m, list #) |
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284 | "USAGE: sysNewton (n,defset,t,q,m,[tr]); n,t,q,m,tr int, defset is list int's |
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285 | @format |
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286 | - n is length, |
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287 | - defset is the defining set, |
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288 | - t is the number of errors, |
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289 | - q is basefield size, |
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290 | - m is degree extension of the splitting field, |
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291 | - if tr>0 it indicates that Newton identities in triangular |
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292 | form should be constructed |
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293 | @end format |
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294 | RETURN: the ring to work with the generalized Newton identities (in |
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295 | triangular form if applicable) containing the ideal with name 'newton' |
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296 | THEORY: Based on 'defset' of the given cyclic code, the procedure constructs |
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297 | the corresponding ideal 'newton' with the generalized Newton |
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298 | identities. With its help one can solve the decoding problem. For |
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299 | basics of the method @ref{Generalized Newton identities}. |
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300 | SEE ALSO: sysCRHT, sysBin |
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301 | EXAMPLE: example sysNewton; shows an example |
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302 | " |
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303 | { |
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304 | string s="ring @newton=("+string(q)+",a),("; |
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305 | int i,j; |
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306 | int flag; |
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307 | int tr; |
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308 | |
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309 | if (size(#)>0) |
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310 | { |
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311 | tr=#[1]; |
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312 | } |
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313 | |
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314 | for (i=n; i>=1; i--) |
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315 | { |
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316 | for (j=1; j<=size(defset); j++) |
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317 | { |
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318 | flag=1; |
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319 | if (i==defset[j]) |
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320 | { |
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321 | flag=0; |
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322 | break; |
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323 | } |
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324 | } |
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325 | if (flag) |
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326 | { |
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327 | s=s+"S("+string(i)+"),"; |
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328 | } |
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329 | } |
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330 | s=s+"sigma(1.."+string(t)+"),"; |
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331 | for (i=size(defset); i>=2; i--) |
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332 | { |
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333 | s=s+"S("+string(defset[i])+"),"; |
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334 | } |
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335 | s=s+"S("+string(defset[1])+")),lp;"; |
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336 | |
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337 | execute(s); |
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338 | |
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339 | ideal newton; |
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340 | poly sum; |
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341 | |
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342 | |
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343 | //------------ generate generalized Newton identities ---------- |
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344 | if (tr) |
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345 | { |
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346 | for (i=1; i<=t; i++) |
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347 | { |
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348 | sum=0; |
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349 | for (j=1; j<=i-1; j++) |
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350 | { |
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351 | sum=sum+sigma(j)*S(i-j); |
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352 | } |
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353 | newton=newton,S(i)+sum+number(i)*sigma(i); |
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354 | } |
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355 | } else |
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356 | { |
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357 | for (i=1; i<=t; i++) |
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358 | { |
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359 | sum=0; |
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360 | for (j=1; j<=t; j++) |
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361 | { |
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362 | sum=sum+sigma(j)*S(mod_(i-j,n)); |
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363 | } |
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364 | newton=newton,S(i)+sum; |
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365 | } |
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366 | } |
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367 | for (i=1; i<=n-t; i++) |
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368 | { |
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369 | sum=0; |
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370 | for (j=1; j<=t; j++) |
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371 | { |
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372 | sum=sum+sigma(j)*S(t+i-j); |
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373 | } |
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374 | newton=newton,S(t+i)+sum; |
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375 | } |
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376 | |
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377 | //----------- add field equations on sigma's -------------- |
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378 | for (i=1; i<=t; i++) |
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379 | { |
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380 | newton=newton,sigma(i)^(q^m)-sigma(i); |
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381 | } |
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382 | |
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383 | //----------- add conjugacy relations ------------------ |
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384 | for (i=1; i<=n; i++) |
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385 | { |
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386 | newton=newton,S(i)^q-S(mod_(q*i,n)); |
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387 | } |
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388 | newton=simplify(newton,2); |
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389 | export newton; |
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390 | return(@newton); |
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391 | } |
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392 | example |
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393 | { |
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394 | "EXAMPLE:"; echo = 2; |
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395 | // Newton identities for a binary 3-error-correcting cyclic code of |
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396 | //length 31 with defining set (1,5,7) |
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397 | |
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398 | int n=31; // length |
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399 | list defset=1,5,7; //defining set |
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400 | int t=3; // number of errors |
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401 | int q=2; // basefield size |
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402 | int m=5; // degree extension of the splitting field |
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403 | int tr=1; // indicator of triangular form of Newton identities |
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404 | |
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405 | |
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406 | def A=sysNewton(n,defset,t,q,m); |
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407 | setring A; |
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408 | A; // shows the ring we are working in |
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409 | print(newton); // generalized Newton identities |
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410 | |
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411 | //=============================== |
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412 | A=sysNewton(n,defset,t,q,m,tr); |
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413 | setring A; |
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414 | print(newton); // generalized Newton identities in triangular form |
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415 | } |
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416 | |
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417 | /////////////////////////////////////////////////////////////////////////////// |
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418 | // forms a list of special combinations needed for computation of Waring's |
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419 | //function |
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420 | static proc combinations_sum (int m, int n) |
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421 | { |
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422 | list result; |
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423 | list comb=combinations(m-1,n+m-1); |
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424 | int i,j,flag,count; |
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425 | list interm=comb; |
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426 | for (i=1; i<=size(comb); i++) |
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427 | { |
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428 | interm[i][1]=comb[i][1]-1; |
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429 | for (j=2; j<=m-1; j++) |
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430 | { |
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431 | interm[i][j]=comb[i][j]-comb[i][j-1]-1; |
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432 | } |
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433 | interm[i][m]=n+m-comb[i][m-1]-1; |
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434 | flag=1; |
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435 | count=2; |
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436 | while ((flag)&&(count<=m)) |
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437 | { |
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438 | if (interm[i][count] mod count != 0) {flag=0;} |
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439 | count++; |
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440 | } |
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441 | if (flag) |
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442 | { |
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443 | for (j=2; j<=m; j++) |
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444 | { |
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445 | interm[i][j]=interm[i][j] div j; |
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446 | } |
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447 | result[size(result)+1]=interm[i]; |
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448 | } |
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449 | } |
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450 | return(result); |
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451 | } |
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452 | |
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453 | /////////////////////////////////////////////////////////////////////////////// |
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454 | //if n=q^e*m, m and n are coprime, then return e |
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455 | static proc exp_count (int n, int q) |
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456 | { |
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457 | int flag=1; |
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458 | int result=0; |
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459 | while(flag) |
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460 | { |
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461 | if (n mod q != 0) {flag=0;} |
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462 | else {n=n div q; result++;} |
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463 | } |
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464 | return(result); |
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465 | } |
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466 | |
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467 | /////////////////////////////////////////////////////////////////////////////// |
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468 | |
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469 | |
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470 | proc sysBin (int v, list Q, int n, list #) |
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471 | "USAGE: sysBin (v,Q,n,[odd]); v,n,odd are int, Q is list of int's |
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472 | @format |
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473 | - v a number if errors, |
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474 | - Q is a defining set of the code, |
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475 | - n the length, |
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476 | - odd is an additional parameter: if |
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477 | set to 1, then the defining set is enlarged by odd elements, |
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478 | which are 2^(some power)*(some elment in the def.set) mod n |
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479 | @end format |
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480 | RETURN: the ring with the resulting system called 'bin' |
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481 | THEORY: Based on Q of the given cyclic code, the procedure constructs |
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482 | the corresponding ideal 'bin' with the use of the Waring function. |
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483 | With its help one can solve the decoding problem. |
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484 | For basics of the method @ref{Generalized Newton identities}. |
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485 | SEE ALSO: sysNewton, sysCRHT |
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486 | EXAMPLE: example sysBin; shows an example |
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487 | " |
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488 | { |
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489 | int odd; |
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490 | if (size(#)>0) |
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491 | { |
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492 | odd=#[1]; |
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493 | } |
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494 | |
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495 | //ring r=2,(sigma(1..v),S(1..n)),(lp(v),dp(n)); |
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496 | ring r=2,(S(1..n),sigma(1..v)),lp; |
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497 | list cyclot; |
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498 | ideal result; |
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499 | int i,j,k,s; |
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500 | list comb; |
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501 | poly sum_, mon; |
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502 | int count1, count2, upper, coef_, flag, gener; |
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503 | list Q_update; |
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504 | if (odd==1) |
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505 | { |
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506 | for (i=1; i<=n; i++) |
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507 | { |
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508 | cyclot[i]=0; |
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509 | } |
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510 | for (i=1; i<=size(Q); i++) |
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511 | { |
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512 | flag=1; |
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513 | gener=Q[i]; |
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514 | while(flag) |
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515 | { |
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516 | cyclot[gener]=1; |
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517 | gener=2*gener mod n; |
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518 | if (gener == Q[i]) {flag=0;} |
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519 | } |
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520 | } |
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521 | for (i=1; i<=n; i++) |
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522 | { |
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523 | if ((cyclot[i] == 1)&&(i mod 2 == 1)) {Q_update[size(Q_update)+1]=i;} |
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524 | } |
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525 | } |
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526 | else |
---|
527 | { |
---|
528 | Q_update=Q; |
---|
529 | } |
---|
530 | |
---|
531 | //---- form polynomials for the Bin system via Waring's function --------- |
---|
532 | for (i=1; i<=size(Q_update); i++) |
---|
533 | { |
---|
534 | comb=combinations_sum(v,Q_update[i]); |
---|
535 | sum_=0; |
---|
536 | for (k=1; k<=size(comb); k++) |
---|
537 | { |
---|
538 | upper=0; |
---|
539 | for (j=1; j<=v; j++) |
---|
540 | { |
---|
541 | upper=upper+comb[k][j]; |
---|
542 | } |
---|
543 | count1=0; |
---|
544 | for (j=2; j<=upper-1; j++) |
---|
545 | { |
---|
546 | count1=count1+exp_count(j,2); |
---|
547 | } |
---|
548 | count1=count1+exp_count(Q_update[i],2); |
---|
549 | count2=0; |
---|
550 | for (j=1; j<=v; j++) |
---|
551 | { |
---|
552 | for (s=2; s<=comb[k][j]; s++) |
---|
553 | { |
---|
554 | count2=count2+exp_count(s,2); |
---|
555 | } |
---|
556 | } |
---|
557 | if (count1<count2) {print("ERRORsysBin");} |
---|
558 | if (count1>count2) {coef_=0;} |
---|
559 | if (count1 == count2) {coef_=1;} |
---|
560 | mon=1; |
---|
561 | for (j=1; j<=v; j++) |
---|
562 | { |
---|
563 | mon=mon*sigma(j)^(comb[k][j]); |
---|
564 | } |
---|
565 | sum_=sum_+coef_*mon; |
---|
566 | } |
---|
567 | result=result,S(Q_update[i])-sum_; |
---|
568 | } |
---|
569 | ideal bin=simplify(result,2); |
---|
570 | export bin; |
---|
571 | return(r); |
---|
572 | } |
---|
573 | example |
---|
574 | { |
---|
575 | "EXAMPLE:"; echo = 2; |
---|
576 | // [31,16,7] quadratic residue code |
---|
577 | list l=1,5,7,9,19,25; |
---|
578 | // we do not need even synromes here |
---|
579 | def A=sysBin(3,l,31); |
---|
580 | setring A; |
---|
581 | print(bin); |
---|
582 | } |
---|
583 | |
---|
584 | /////////////////////////////////////////////////////////////////////////////// |
---|
585 | |
---|
586 | proc encode (matrix x, matrix g) |
---|
587 | "USAGE: encode (x, g); x a row vector (message), and g a generator matrix |
---|
588 | RETURN: corresponding codeword |
---|
589 | EXAMPLE: example encode; shows an example |
---|
590 | " |
---|
591 | { |
---|
592 | if (nrows(x)>1) {print("ERRORencode1!");} |
---|
593 | if (ncols(x)!=nrows(g)) {print("ERRORencode2!");} |
---|
594 | return(x*g); |
---|
595 | } |
---|
596 | example |
---|
597 | { |
---|
598 | "EXAMPLE:"; echo = 2; |
---|
599 | ring r=2,x,dp; |
---|
600 | matrix x[1][4]=1,0,1,0; |
---|
601 | matrix g[4][7]=1,0,0,0,0,1,1, |
---|
602 | 0,1,0,0,1,0,1, |
---|
603 | 0,0,1,0,1,1,1, |
---|
604 | 0,0,0,1,1,1,0; |
---|
605 | //encode x with the generator matrix g |
---|
606 | print(encode(x,g)); |
---|
607 | } |
---|
608 | |
---|
609 | /////////////////////////////////////////////////////////////////////////////// |
---|
610 | |
---|
611 | proc syndrome (matrix h, matrix c) |
---|
612 | "USAGE: syndrome (h, c); h a check matrix, c a row vector (codeword) |
---|
613 | RETURN: corresponding syndrome |
---|
614 | EXAMPLE: example syndrome; shows an example |
---|
615 | " |
---|
616 | { |
---|
617 | if (nrows(c)>1) {print("ERRORsyndrome1!");} |
---|
618 | if (ncols(c)!=ncols(h)) {print("ERRORsyndrome2!");} |
---|
619 | return(h*transpose(c)); |
---|
620 | } |
---|
621 | example |
---|
622 | { |
---|
623 | "EXAMPLE:"; echo = 2; |
---|
624 | ring r=2,x,dp; |
---|
625 | matrix x[1][4]=1,0,1,0; |
---|
626 | matrix g[4][7]=1,0,0,0,0,1,1, |
---|
627 | 0,1,0,0,1,0,1, |
---|
628 | 0,0,1,0,1,1,1, |
---|
629 | 0,0,0,1,1,1,0; |
---|
630 | //encode x with the generator matrix g |
---|
631 | matrix c=encode(x,g); |
---|
632 | // disturb |
---|
633 | c[1,3]=0; |
---|
634 | //compute syndrome |
---|
635 | //corresponding check matrix |
---|
636 | matrix check[3][7]=1,0,0,1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,1; |
---|
637 | print(syndrome(check,c)); |
---|
638 | c[1,3]=1; |
---|
639 | //now c is a codeword |
---|
640 | print(syndrome(check,c)); |
---|
641 | } |
---|
642 | |
---|
643 | /////////////////////////////////////////////////////////////////////////////// |
---|
644 | // (coordinatewise) star product of two vectors |
---|
645 | static proc star(matrix m, int i, int j) |
---|
646 | { |
---|
647 | matrix result[ncols(m)][1]; |
---|
648 | for (int k=1; k<=ncols(m); k++) |
---|
649 | { |
---|
650 | result[k,1]=m[i,k]*m[j,k]; |
---|
651 | } |
---|
652 | return(result); |
---|
653 | } |
---|
654 | |
---|
655 | /////////////////////////////////////////////////////////////////////////////// |
---|
656 | |
---|
657 | proc sysQE(matrix check, matrix y, int t, list #) |
---|
658 | "USAGE: sysQE(check,y,t,[fieldeq,formal]);check,y matrix;t,fieldeq,formal int |
---|
659 | @format |
---|
660 | - check is a parity check matrix of the code |
---|
661 | - y is a received word, |
---|
662 | - t the number of errors to be corrected, |
---|
663 | - if fieldeq=1, then field equations are added, |
---|
664 | - if formal=0, field equations on (known) syndrome variables |
---|
665 | are not added, in order to add them (note that the exponent should |
---|
666 | be equal to the number of elements in the INITIAL alphabet) one |
---|
667 | needs to set formal>0 for the exponent |
---|
668 | @end format |
---|
669 | RETURN: the ring to work with together with the resulting system called 'qe' |
---|
670 | THEORY: Based on 'check' of the given linear code, the procedure constructs |
---|
671 | the corresponding ideal that gives an opportunity to compute |
---|
672 | unknown syndrome of the received word y. After computing the unknown |
---|
673 | syndromes one is able to solve the decoding problem. |
---|
674 | For basics of the method @ref{Decoding method based on quadratic equations}. |
---|
675 | SEE ALSO: sysFL |
---|
676 | EXAMPLE: example sysQE; shows an example |
---|
677 | " |
---|
678 | { |
---|
679 | int fieldeq; |
---|
680 | int formal; |
---|
681 | if (size(#)>0) |
---|
682 | { |
---|
683 | fieldeq=#[1]; |
---|
684 | } |
---|
685 | if (size(#)>1) |
---|
686 | { |
---|
687 | formal=#[2]; |
---|
688 | } |
---|
689 | |
---|
690 | def br=basering; |
---|
691 | list rl=ringlist(br); |
---|
692 | |
---|
693 | int red=nrows(check); |
---|
694 | int n=ncols(check); |
---|
695 | int q=rl[1][1]; |
---|
696 | |
---|
697 | if (formal==0) |
---|
698 | { |
---|
699 | ring work=(q,a),(V(1..t),U(1..n)),dp; |
---|
700 | } else |
---|
701 | { |
---|
702 | ring work=(q,a),(V(1..t),U(1..n),s(1..red)),(dp(t),lp(n),dp(red)); |
---|
703 | } |
---|
704 | |
---|
705 | matrix check=imap(br,check); |
---|
706 | matrix y=imap(br,y); |
---|
707 | |
---|
708 | matrix h_full=genMDSMat(n,a); |
---|
709 | matrix h=submat(h_full,1..red,1..n); |
---|
710 | if (nrows(y)!=1) {print("ERROR1Pell");} |
---|
711 | if (ncols(y)!=n) {print("ERROR2Pell");} |
---|
712 | |
---|
713 | ideal result; |
---|
714 | |
---|
715 | list c; |
---|
716 | list a; |
---|
717 | list tmp,tmp2; |
---|
718 | int i,j,l,k; |
---|
719 | number sum,prod,sig; |
---|
720 | poly sum1,sum2,sum3; |
---|
721 | for (i=1; i<=n; i++) |
---|
722 | { |
---|
723 | c[i]=tmp; |
---|
724 | } |
---|
725 | |
---|
726 | matrix transf=inverse(transpose(h_full)); |
---|
727 | |
---|
728 | //------ expression matrix of check vectors w.r.t. the MDS basis ----------- |
---|
729 | for (i=1; i<=red ; i++) |
---|
730 | { |
---|
731 | a[i]=transpose(submat(check,i..i,1..n)); |
---|
732 | a[i]=transf*a[i]; |
---|
733 | } |
---|
734 | |
---|
735 | //----------- compute the structure constants ------------------------ |
---|
736 | matrix te[n][1]; |
---|
737 | for (i=1; i<=n; i++) |
---|
738 | { |
---|
739 | for (j=1; j<=t+1; j++) |
---|
740 | { |
---|
741 | if ((j<i)&&(i<=t+1)) {c[i][j]=c[j][i];} |
---|
742 | else |
---|
743 | { |
---|
744 | if (i+j<=n+1) |
---|
745 | { |
---|
746 | c[i][j]=te; |
---|
747 | c[i][j][i+j-1,1]=1; |
---|
748 | } |
---|
749 | else |
---|
750 | { |
---|
751 | c[i][j]=star(h_full,i,j); |
---|
752 | c[i][j]=transf*c[i][j]; |
---|
753 | } |
---|
754 | } |
---|
755 | } |
---|
756 | } |
---|
757 | |
---|
758 | |
---|
759 | if (formal==0) |
---|
760 | { |
---|
761 | matrix s[red][1]=syndrome(check,y); |
---|
762 | for (j=1; j<=red; j++) |
---|
763 | { |
---|
764 | sum1=0; |
---|
765 | for (l=1; l<=n; l++) |
---|
766 | { |
---|
767 | sum1=sum1+a[j][l,1]*U(l); |
---|
768 | } |
---|
769 | result=result,sum1-s[j,1]; |
---|
770 | } |
---|
771 | } else |
---|
772 | { |
---|
773 | for (j=1; j<=red; j++) |
---|
774 | { |
---|
775 | sum1=0; |
---|
776 | for (l=1; l<=n; l++) |
---|
777 | { |
---|
778 | sum1=sum1+a[j][l,1]*U(l); |
---|
779 | } |
---|
780 | result=result,sum1-s(j); |
---|
781 | } |
---|
782 | for (j=1; j<=red; j++) |
---|
783 | { |
---|
784 | result=result,s(j)^(formal)-s(j); |
---|
785 | } |
---|
786 | } |
---|
787 | if (fieldeq) |
---|
788 | { |
---|
789 | for (i=1; i<=n; i++) |
---|
790 | { |
---|
791 | result=result,U(i)^q-U(i); |
---|
792 | } |
---|
793 | for (j=1; j<=t; j++) |
---|
794 | { |
---|
795 | result=result,V(j)^q-V(j); |
---|
796 | } |
---|
797 | } |
---|
798 | |
---|
799 | //----- form the quadratic equations according to the theory ----------- |
---|
800 | for (i=1; i<=n; i++) |
---|
801 | { |
---|
802 | sum1=0; |
---|
803 | for (j=1; j<=t; j++) |
---|
804 | { |
---|
805 | sum2=0; |
---|
806 | for (l=1; l<=n; l++) |
---|
807 | { |
---|
808 | sum2=sum2+c[i][j][l,1]*U(l); |
---|
809 | } |
---|
810 | sum1=sum1+sum2*V(j); |
---|
811 | } |
---|
812 | sum3=0; |
---|
813 | for (l=1; l<=n; l++) |
---|
814 | { |
---|
815 | sum3=sum3+c[i][t+1][l,1]*U(l); |
---|
816 | } |
---|
817 | result=result,sum1-sum3; |
---|
818 | } |
---|
819 | |
---|
820 | result=simplify(result,2); |
---|
821 | |
---|
822 | ideal qe=result; |
---|
823 | export qe; |
---|
824 | return(work); |
---|
825 | } |
---|
826 | example |
---|
827 | { |
---|
828 | "EXAMPLE:"; echo = 2; |
---|
829 | intvec v = option(get); |
---|
830 | |
---|
831 | //correct 2 errors in [7,3] 8-ary code RS code |
---|
832 | int t=2; int q=8; int n=7; int redun=4; |
---|
833 | ring r=(q,a),x,dp; |
---|
834 | matrix h_full=genMDSMat(n,a); |
---|
835 | matrix h=submat(h_full,1..redun,1..n); |
---|
836 | matrix g=dual_code(h); |
---|
837 | matrix x[1][3]=0,0,1,0; |
---|
838 | matrix y[1][7]=encode(x,g); |
---|
839 | |
---|
840 | //disturb with 2 errors |
---|
841 | matrix rec[1][7]=errorInsert(y,list(2,4),list(1,a)); |
---|
842 | |
---|
843 | //generate the system |
---|
844 | def A=sysQE(h,rec,t); |
---|
845 | setring A; |
---|
846 | print(qe); |
---|
847 | |
---|
848 | //let us decode |
---|
849 | option(redSB); |
---|
850 | ideal sys_qe=std(qe); |
---|
851 | print(sys_qe); |
---|
852 | |
---|
853 | option(set,v); |
---|
854 | } |
---|
855 | |
---|
856 | /////////////////////////////////////////////////////////////////////////////// |
---|
857 | |
---|
858 | proc errorInsert(matrix y, list pos, list val) |
---|
859 | "USAGE: errorInsert(y,pos,val); y is matrix, pos,val are list of int's |
---|
860 | @format |
---|
861 | - y is a (code) word, |
---|
862 | - pos = positions where errors occured, |
---|
863 | - val = their corresponding values |
---|
864 | @end format |
---|
865 | RETURN: corresponding received word |
---|
866 | EXAMPLE: example errorInsert; shows an example |
---|
867 | " |
---|
868 | { |
---|
869 | matrix result[1][ncols(y)]=y; |
---|
870 | if (size(pos)!=size(val)) {print("ERRORerror");} |
---|
871 | for (int i=1; i<=size(pos); i++) |
---|
872 | { |
---|
873 | result[1,pos[i]]=y[1,pos[i]]+val[i]; |
---|
874 | } |
---|
875 | return(result); |
---|
876 | } |
---|
877 | example |
---|
878 | { |
---|
879 | "EXAMPLE:"; echo = 2; |
---|
880 | //correct 2 errors in [7,3] 8-ary code RS code |
---|
881 | int t=2; int q=8; int n=7; int redun=4; |
---|
882 | ring r=(q,a),x,dp; |
---|
883 | matrix h_full=genMDSMat(n,a); |
---|
884 | matrix h=submat(h_full,1..redun,1..n); |
---|
885 | matrix g=dual_code(h); |
---|
886 | matrix x[1][3]=0,0,1,0; |
---|
887 | matrix y[1][7]=encode(x,g); |
---|
888 | print(y); |
---|
889 | |
---|
890 | //disturb with 2 errors |
---|
891 | matrix rec[1][7]=errorInsert(y,list(2,4),list(1,a)); |
---|
892 | print(rec); |
---|
893 | print(rec-y); |
---|
894 | } |
---|
895 | |
---|
896 | /////////////////////////////////////////////////////////////////////////////// |
---|
897 | |
---|
898 | proc errorRand(matrix y, int num, int e) |
---|
899 | "USAGE: errorRand(y, num, e); y is matrix, num,e are int |
---|
900 | @format |
---|
901 | - y is a (code) word, |
---|
902 | - num is the number of errors, |
---|
903 | - e is an extension degree (if one wants values to be from GF(p^e)) |
---|
904 | @end format |
---|
905 | RETURN: corresponding received word |
---|
906 | EXAMPLE: example errorRand; shows an example |
---|
907 | " |
---|
908 | { |
---|
909 | matrix result[1][ncols(y)]=y; |
---|
910 | int i,j, flag, temp; |
---|
911 | list pos, val; |
---|
912 | matrix tempnum; |
---|
913 | |
---|
914 | for (i=1; i<=num; i++) |
---|
915 | { |
---|
916 | while(1) |
---|
917 | { |
---|
918 | temp=random(1,ncols(y)); |
---|
919 | flag=1; |
---|
920 | for (j=1; j<=size(pos); j++) |
---|
921 | { |
---|
922 | if (temp==pos[j]) {flag=0;} |
---|
923 | } |
---|
924 | if (flag) {pos[i]=temp;break;} |
---|
925 | } |
---|
926 | } |
---|
927 | |
---|
928 | for (i=1; i<=num; i++) |
---|
929 | { |
---|
930 | flag=1; |
---|
931 | while(flag) |
---|
932 | { |
---|
933 | tempnum=randomvector(1,e); |
---|
934 | if (tempnum!=0) {flag=0;} |
---|
935 | } |
---|
936 | val[i]=tempnum; |
---|
937 | } |
---|
938 | |
---|
939 | for (i=1; i<=size(pos); i++) |
---|
940 | { |
---|
941 | result[1,pos[i]]=y[1,pos[i]]+val[i]; |
---|
942 | } |
---|
943 | return(result); |
---|
944 | } |
---|
945 | example |
---|
946 | { |
---|
947 | "EXAMPLE:"; echo = 2; |
---|
948 | //correct 2 errors in [7,3] 8-ary code RS code |
---|
949 | int t=2; int q=8; int n=7; int redun=4; |
---|
950 | ring r=(q,a),x,dp; |
---|
951 | matrix h_full=genMDSMat(n,a); |
---|
952 | matrix h=submat(h_full,1..redun,1..n); |
---|
953 | matrix g=dual_code(h); |
---|
954 | matrix x[1][3]=0,0,1,0; |
---|
955 | matrix y[1][7]=encode(x,g); |
---|
956 | |
---|
957 | //disturb with 2 random errors |
---|
958 | matrix rec[1][7]=errorRand(y,2,3); |
---|
959 | print(rec); |
---|
960 | print(rec-y); |
---|
961 | } |
---|
962 | |
---|
963 | /////////////////////////////////////////////////////////////////////////////// |
---|
964 | |
---|
965 | proc randomCheck(int m, int n, int e) |
---|
966 | "USAGE: randomCheck(m, n, e); m,n,e are int |
---|
967 | @format |
---|
968 | - m x n are dimensions of the matrix, |
---|
969 | - e is an extension degree (if one wants values to be from GF(p^e)) |
---|
970 | @end format |
---|
971 | RETURN: random check matrix |
---|
972 | EXAMPLE: example randomCheck; shows an example |
---|
973 | " |
---|
974 | { |
---|
975 | matrix result[m][n]; |
---|
976 | matrix rand[m][n-m]; |
---|
977 | int i,j; |
---|
978 | matrix temp; |
---|
979 | for (i=1; i<=m; i++) |
---|
980 | { |
---|
981 | temp=randomvector(n-m,e); |
---|
982 | for (j=1; j<=n-m; j++) |
---|
983 | { |
---|
984 | rand[i,j]=temp[j,1]; |
---|
985 | } |
---|
986 | } |
---|
987 | result=concat(rand,unitmat(m)); |
---|
988 | return(result); |
---|
989 | } |
---|
990 | example |
---|
991 | { |
---|
992 | "EXAMPLE:"; echo = 2; |
---|
993 | int redun=5; int n=15; |
---|
994 | ring r=2,x,dp; |
---|
995 | |
---|
996 | //generate random check matrix for a [15,5] binary code |
---|
997 | matrix h=randomCheck(redun,n,1); |
---|
998 | print(h); |
---|
999 | |
---|
1000 | //corresponding generator matrix |
---|
1001 | matrix g=dual_code(h); |
---|
1002 | print(g); |
---|
1003 | } |
---|
1004 | |
---|
1005 | /////////////////////////////////////////////////////////////////////////////// |
---|
1006 | |
---|
1007 | proc genMDSMat(int n, number a) |
---|
1008 | "USAGE: genMDSMat(n, a); n is int, a is number |
---|
1009 | @format |
---|
1010 | - n x n are dimensions of the MDS matrix, |
---|
1011 | - a is a primitive element of the field. |
---|
1012 | @end format |
---|
1013 | NOTE: An MDS matrix is constructed in the following way. We take 'a' to be a |
---|
1014 | generator of the multiplicative group of the field. Then we construct |
---|
1015 | the Vandermonde matrix with this 'a'. |
---|
1016 | ASSUME: extension field should already be defined |
---|
1017 | RETURN: a matrix with the MDS property. |
---|
1018 | SEE ALSO: Decoding method based on quadratic equations |
---|
1019 | EXAMPLE: example genMDSMat; shows an example |
---|
1020 | " |
---|
1021 | { |
---|
1022 | int i,j; |
---|
1023 | matrix result[n][n]; |
---|
1024 | for (i=0; i<=n-1; i++) |
---|
1025 | { |
---|
1026 | for (j=0; j<=n-1; j++) |
---|
1027 | { |
---|
1028 | result[j+1,i+1]=(a^i)^j; |
---|
1029 | } |
---|
1030 | } |
---|
1031 | return(result); |
---|
1032 | } |
---|
1033 | example |
---|
1034 | { |
---|
1035 | "EXAMPLE:"; echo = 2; |
---|
1036 | int q=16; int n=15; |
---|
1037 | ring r=(q,a),x,dp; |
---|
1038 | |
---|
1039 | //generate an MDS (Vandermonde) matrix |
---|
1040 | matrix h_full=genMDSMat(n,a); |
---|
1041 | print(h_full); |
---|
1042 | } |
---|
1043 | |
---|
1044 | /////////////////////////////////////////////////////////////////////////////// |
---|
1045 | |
---|
1046 | |
---|
1047 | proc mindist (matrix check) |
---|
1048 | "USAGE: mindist (check, q); check matrix, q int |
---|
1049 | @format |
---|
1050 | - check is a check matrix, |
---|
1051 | - q is the field size |
---|
1052 | @end format |
---|
1053 | RETURN: minimum distance of the code |
---|
1054 | EXAMPLE: example mindist; shows an example |
---|
1055 | " |
---|
1056 | { |
---|
1057 | intvec vopt = option(get); |
---|
1058 | |
---|
1059 | int n=ncols(check); int redun=nrows(check); int t=redun+1; |
---|
1060 | |
---|
1061 | def br=basering; |
---|
1062 | list rl=ringlist(br); |
---|
1063 | int q=rl[1][1]; |
---|
1064 | |
---|
1065 | ring work=(q,a),(V(1..t),U(1..n)),dp; |
---|
1066 | matrix check=imap(br,check); |
---|
1067 | |
---|
1068 | ideal temp; |
---|
1069 | int count=1; |
---|
1070 | int flag=1; |
---|
1071 | int flag2; |
---|
1072 | int i; |
---|
1073 | matrix z[1][n]; |
---|
1074 | option(redSB); |
---|
1075 | def A=sysQE(check,z,count); |
---|
1076 | |
---|
1077 | //proceed with solving the system w.r.t zero vector until some solutions |
---|
1078 | //are found |
---|
1079 | while (flag) |
---|
1080 | { |
---|
1081 | A=sysQE(check,z,count); |
---|
1082 | setring A; |
---|
1083 | ideal temp=qe; |
---|
1084 | temp=std(temp); |
---|
1085 | flag2=1; |
---|
1086 | setring work; |
---|
1087 | temp=imap(A,temp); |
---|
1088 | for (i=1; i<=n; i++) |
---|
1089 | { |
---|
1090 | if |
---|
1091 | (temp[i]!=U(n-i+1)) |
---|
1092 | { |
---|
1093 | flag2=0; |
---|
1094 | } |
---|
1095 | } |
---|
1096 | if (!flag2) |
---|
1097 | { |
---|
1098 | flag=0; |
---|
1099 | } |
---|
1100 | else |
---|
1101 | { |
---|
1102 | count++; |
---|
1103 | } |
---|
1104 | } |
---|
1105 | int result=count; |
---|
1106 | |
---|
1107 | option(set,vopt); |
---|
1108 | return(result); |
---|
1109 | } |
---|
1110 | example |
---|
1111 | { |
---|
1112 | "EXAMPLE:"; echo = 2; |
---|
1113 | //determine a minimum distance for a [7,3] binary code |
---|
1114 | int q=8; int n=7; int redun=4; int t=redun+1; |
---|
1115 | ring r=(q,a),x,dp; |
---|
1116 | |
---|
1117 | //generate random check matrix |
---|
1118 | matrix h=randomCheck(redun,n,1); |
---|
1119 | print(h); |
---|
1120 | int l=mindist(h); |
---|
1121 | l; |
---|
1122 | } |
---|
1123 | |
---|
1124 | /////////////////////////////////////////////////////////////////////////////// |
---|
1125 | |
---|
1126 | proc decode(matrix check, matrix rec) |
---|
1127 | "USAGE: decode(check, rec, t); check, rec matrix, t int |
---|
1128 | @format |
---|
1129 | - check is the check matrix of the code, |
---|
1130 | - rec is a received word, |
---|
1131 | - t is an upper bound for the number of errors one wants to correct |
---|
1132 | @end format |
---|
1133 | NOTE: The method described in @ref{Decoding method based on quadratic equations} |
---|
1134 | is used for decoding. |
---|
1135 | ASSUME: Errors in rec should be correctable, otherwise the output is |
---|
1136 | unpredictable |
---|
1137 | RETURN: a codeword that is closest to rec |
---|
1138 | EXAMPLE: example decode; shows an example |
---|
1139 | " |
---|
1140 | { |
---|
1141 | intvec vopt = option(get); |
---|
1142 | |
---|
1143 | def br=basering; |
---|
1144 | int n=ncols(check); |
---|
1145 | |
---|
1146 | int count=1; |
---|
1147 | def A=sysQE(check,rec,count); |
---|
1148 | while(1) |
---|
1149 | { |
---|
1150 | A=sysQE(check,rec,count); |
---|
1151 | setring A; |
---|
1152 | matrix h_full=genMDSMat(n,a); |
---|
1153 | matrix rec=imap(br,rec); |
---|
1154 | option(redSB); |
---|
1155 | ideal qe_red=std(qe); |
---|
1156 | if (qe_red[1]!=1) |
---|
1157 | { |
---|
1158 | break; |
---|
1159 | } |
---|
1160 | else |
---|
1161 | { |
---|
1162 | count++; |
---|
1163 | } |
---|
1164 | setring br; |
---|
1165 | } |
---|
1166 | |
---|
1167 | setring A; |
---|
1168 | |
---|
1169 | //obtain a codeword |
---|
1170 | //this works only if our code is indeed can correct these errors |
---|
1171 | matrix syn[n][1]; |
---|
1172 | for (int i=1; i<=n; i++) |
---|
1173 | { |
---|
1174 | syn[i,1]=-qe_red[n-i+1]+lead(qe_red[n-i+1]); |
---|
1175 | } |
---|
1176 | |
---|
1177 | matrix real_syn=inverse(h_full)*syn; |
---|
1178 | setring br; |
---|
1179 | matrix real_syn=imap(A,real_syn); |
---|
1180 | |
---|
1181 | option(set,vopt); |
---|
1182 | return(rec-transpose(real_syn)); |
---|
1183 | } |
---|
1184 | example |
---|
1185 | { |
---|
1186 | "EXAMPLE:"; echo = 2; |
---|
1187 | //correct 1 error in [15,7] binary code |
---|
1188 | int t=1; int q=16; int n=15; int redun=10; |
---|
1189 | ring r=(q,a),x,dp; |
---|
1190 | |
---|
1191 | //generate random check matrix |
---|
1192 | matrix h=randomCheck(redun,n,1); |
---|
1193 | matrix g=dual_code(h); |
---|
1194 | matrix x[1][n-redun]=0,0,1,0,1,0,1; |
---|
1195 | matrix y[1][n]=encode(x,g); |
---|
1196 | print(y); |
---|
1197 | |
---|
1198 | // find out the minimum distance of the code |
---|
1199 | list l=mindist(h); |
---|
1200 | |
---|
1201 | //disturb with errors |
---|
1202 | "Correct ",(l[1]-1) div 2," errors"; |
---|
1203 | matrix rec[1][n]=errorRand(y,(l[1]-1) div 2,1); |
---|
1204 | print(rec); |
---|
1205 | |
---|
1206 | //let us decode |
---|
1207 | matrix dec_word=decode(h,rec); |
---|
1208 | print(dec_word); |
---|
1209 | } |
---|
1210 | |
---|
1211 | /////////////////////////////////////////////////////////////////////////////// |
---|
1212 | |
---|
1213 | |
---|
1214 | proc decodeRandom(int n, int redun, int ncodes, int ntrials, list #) |
---|
1215 | "USAGE: decodeRandom(redun,q,ncodes,ntrials,[e]); all parameters int |
---|
1216 | @format |
---|
1217 | - redun is a redundabcy of a (random) code, |
---|
1218 | - q is the field size, |
---|
1219 | - ncodes is the number of random codes to be processed, |
---|
1220 | - ntrials is the number of received vectors per code to be corrected |
---|
1221 | - If e is given it sets the correction capacity explicitly. It |
---|
1222 | should be used in case one expects some lower bound, |
---|
1223 | otherwise the procedure tries to compute the real minimum distance |
---|
1224 | to find out the error-correction capacity |
---|
1225 | @end format |
---|
1226 | RETURN: nothing; |
---|
1227 | EXAMPLE: example decodeRandom; shows an example |
---|
1228 | " |
---|
1229 | { |
---|
1230 | intvec vopt = option(get); |
---|
1231 | |
---|
1232 | int i,j; |
---|
1233 | matrix h; |
---|
1234 | int dist, t; |
---|
1235 | ideal sys; |
---|
1236 | int tmp; |
---|
1237 | int e; |
---|
1238 | if (size(#)>0) |
---|
1239 | { |
---|
1240 | e=#[1]; |
---|
1241 | } |
---|
1242 | |
---|
1243 | option(redSB); |
---|
1244 | def br=basering; |
---|
1245 | matrix h_full=genMDSMat(n,a); |
---|
1246 | matrix z[1][ncols(h_full)]; |
---|
1247 | |
---|
1248 | //------------------ determine error-correction capacity ------------------- |
---|
1249 | for (i=1; i<=ncodes; i++) |
---|
1250 | { |
---|
1251 | setring br; |
---|
1252 | h=randomCheck(redun,n,1); |
---|
1253 | "check matrix:"; |
---|
1254 | print(h); |
---|
1255 | if (e>0) |
---|
1256 | { |
---|
1257 | t=e; |
---|
1258 | } else { |
---|
1259 | tmp=mindist(h); |
---|
1260 | dist=tmp; |
---|
1261 | printf("d= %p",dist); |
---|
1262 | t=(dist-1) div 2; |
---|
1263 | } |
---|
1264 | |
---|
1265 | //------------- generate the template system ---------------------- |
---|
1266 | def A=sysQE(h,z,t); |
---|
1267 | setring A; |
---|
1268 | matrix word,y,rec; |
---|
1269 | ideal sys2,sys3; |
---|
1270 | matrix h=imap(br,h); |
---|
1271 | matrix g=dual_code(h); |
---|
1272 | ideal sys=qe; |
---|
1273 | print("The system is generated"); |
---|
1274 | |
---|
1275 | //------ modify the template according to every received word -------------- |
---|
1276 | for (j=1; j<=ntrials; j++) |
---|
1277 | { |
---|
1278 | word=randomvector(n-redun,1); |
---|
1279 | y=encode(transpose(word),g); |
---|
1280 | print("Codeword:"); |
---|
1281 | print(y); |
---|
1282 | rec=errorRand(y,t,1); |
---|
1283 | print("Received word:"); |
---|
1284 | print(rec); |
---|
1285 | sys2=add_synd(rec,h,redun,sys); |
---|
1286 | option(redSB); |
---|
1287 | sys3=std(sys2); |
---|
1288 | print("The Groebenr basis of the QE system:"); |
---|
1289 | print(sys3); |
---|
1290 | } |
---|
1291 | kill A; |
---|
1292 | option(set,vopt); |
---|
1293 | } |
---|
1294 | } |
---|
1295 | example |
---|
1296 | { |
---|
1297 | "EXAMPLE:"; echo = 2; |
---|
1298 | int q=32; int n=25; int redun=n-11; int t=redun+1; |
---|
1299 | ring r=(q,a),x,dp; |
---|
1300 | |
---|
1301 | // correct 2 errors in 2 random binary codes, 3 trials each |
---|
1302 | decodeRandom(n,redun,2,3,2); |
---|
1303 | } |
---|
1304 | |
---|
1305 | /////////////////////////////////////////////////////////////////////////////// |
---|
1306 | |
---|
1307 | |
---|
1308 | proc decodeCode(matrix check, int ntrials, list #) |
---|
1309 | "USAGE: decodeCode(check, ntrials, [e]); check matrix, ntrials,e int |
---|
1310 | @format |
---|
1311 | - check is a parity check matrix for the code, |
---|
1312 | - ntrials is the number of received vectors per code to be |
---|
1313 | corrected. |
---|
1314 | - If e is given it sets the correction capacity explicitly. It |
---|
1315 | should be used in case one expects some lower bound, |
---|
1316 | otherwise the procedure tries to compute the real minimum distance |
---|
1317 | to find out the error-correction capacity |
---|
1318 | @end format |
---|
1319 | RETURN: nothing; |
---|
1320 | EXAMPLE: example decodeCode; shows an example |
---|
1321 | " |
---|
1322 | { |
---|
1323 | intvec vopt = option(get); |
---|
1324 | |
---|
1325 | int n=ncols(check); |
---|
1326 | int redun=nrows(check); |
---|
1327 | int i,j; |
---|
1328 | matrix h; |
---|
1329 | int dist, t; |
---|
1330 | ideal sys; |
---|
1331 | int tmp; |
---|
1332 | int e; |
---|
1333 | if (size(#)>0) |
---|
1334 | { |
---|
1335 | e=#[1]; |
---|
1336 | } |
---|
1337 | |
---|
1338 | option(redSB); |
---|
1339 | def br=basering; |
---|
1340 | matrix h_full=genMDSMat(n,a); |
---|
1341 | matrix z[1][ncols(h_full)]; |
---|
1342 | setring br; |
---|
1343 | h=check; |
---|
1344 | "check matrix:"; |
---|
1345 | print(h); |
---|
1346 | |
---|
1347 | //------------------ determine error-correction capacity ------------------- |
---|
1348 | if (e>0) |
---|
1349 | { |
---|
1350 | t=e; |
---|
1351 | } else { |
---|
1352 | tmp=mindist(h); |
---|
1353 | dist=tmp; |
---|
1354 | printf("d= %p",dist); |
---|
1355 | t=(dist-1) div 2; |
---|
1356 | } |
---|
1357 | |
---|
1358 | //------------- generate the template system ---------------------- |
---|
1359 | def A=sysQE(h,z,t); |
---|
1360 | setring A; |
---|
1361 | matrix word,y,rec; |
---|
1362 | ideal sys2,sys3; |
---|
1363 | matrix h=imap(br,h); |
---|
1364 | matrix g=dual_code(h); |
---|
1365 | ideal sys=qe; |
---|
1366 | print("The system is generated"); |
---|
1367 | |
---|
1368 | //--- modify the template according to every received word --------------- |
---|
1369 | for (j=1; j<=ntrials; j++) |
---|
1370 | { |
---|
1371 | word=randomvector(n-redun,1); |
---|
1372 | y=encode(transpose(word),g); |
---|
1373 | print("Codeword:"); |
---|
1374 | print(y); |
---|
1375 | rec=errorRand(y,t,1); |
---|
1376 | print("Received word:"); |
---|
1377 | print(rec); |
---|
1378 | sys2=add_synd(rec,h,redun,sys); |
---|
1379 | option(redSB); |
---|
1380 | sys3=std(sys2); |
---|
1381 | print("Groebner basis of the QE system:"); |
---|
1382 | print(sys3); |
---|
1383 | } |
---|
1384 | |
---|
1385 | option(set,vopt); |
---|
1386 | } |
---|
1387 | example |
---|
1388 | { |
---|
1389 | "EXAMPLE:"; echo = 2; |
---|
1390 | int q=32; int n=25; int redun=n-11; int t=redun+1; |
---|
1391 | ring r=(q,a),x,dp; |
---|
1392 | matrix check=randomCheck(redun,n,1); |
---|
1393 | |
---|
1394 | // correct 2 errors in using the code above, 3 trials |
---|
1395 | decodeCode(check,3,2); |
---|
1396 | } |
---|
1397 | |
---|
1398 | |
---|
1399 | /////////////////////////////////////////////////////////////////////////////// |
---|
1400 | // adding syndrome values to the template system |
---|
1401 | static proc add_synd (matrix rec, matrix check, int redun, ideal sys) |
---|
1402 | { |
---|
1403 | ideal result=sys; |
---|
1404 | matrix s[redun][1]=syndrome(check,rec); |
---|
1405 | for (int i=1; i<=redun; i++) |
---|
1406 | |
---|
1407 | { |
---|
1408 | result[i]=result[i]-s[i,1]; |
---|
1409 | } |
---|
1410 | return(result); |
---|
1411 | } |
---|
1412 | |
---|
1413 | /////////////////////////////////////////////////////////////////////////////// |
---|
1414 | // evaluate a polynomial at a given point |
---|
1415 | static proc ev (poly f, matrix p) |
---|
1416 | { |
---|
1417 | if (ncols(p)>1) {ERROR("not a column vector");}; |
---|
1418 | int m=size(p); |
---|
1419 | poly temp=f; |
---|
1420 | for (int i=1; i<=m; i++) |
---|
1421 | { |
---|
1422 | temp=subst(temp,var(i),p[i,1]); |
---|
1423 | } |
---|
1424 | return(number(temp)); |
---|
1425 | } |
---|
1426 | |
---|
1427 | /////////////////////////////////////////////////////////////////////////////// |
---|
1428 | // return index of an element in the ideal where it does not vanish at the |
---|
1429 | //given point |
---|
1430 | static proc find_index (ideal G, matrix p) |
---|
1431 | { |
---|
1432 | if (ncols(p)>1) {ERROR("not a column vector");}; |
---|
1433 | int i=1; |
---|
1434 | int n=size(G); |
---|
1435 | while(i<=n) |
---|
1436 | { |
---|
1437 | if (ev(G[i],p)!=0) {return(i);} |
---|
1438 | i++; |
---|
1439 | } |
---|
1440 | return(-1); |
---|
1441 | } |
---|
1442 | |
---|
1443 | /////////////////////////////////////////////////////////////////////////////// |
---|
1444 | // convert ideal to list |
---|
1445 | static proc ideal2list (ideal id) |
---|
1446 | { |
---|
1447 | list l; |
---|
1448 | for (int i=1; i<=size(id); i++) |
---|
1449 | { |
---|
1450 | l[i]=id[i]; |
---|
1451 | } |
---|
1452 | return(l); |
---|
1453 | } |
---|
1454 | |
---|
1455 | /////////////////////////////////////////////////////////////////////////////// |
---|
1456 | // convert list to ideal |
---|
1457 | static proc list2ideal (list l) |
---|
1458 | { |
---|
1459 | ideal id; |
---|
1460 | for (int i=1; i<=size(l); i++) |
---|
1461 | { |
---|
1462 | id[i]=l[i]; |
---|
1463 | } |
---|
1464 | return(id); |
---|
1465 | } |
---|
1466 | |
---|
1467 | /////////////////////////////////////////////////////////////////////////////// |
---|
1468 | // check whether given polynomial is divisible by some leading monomial of the |
---|
1469 | //ideal |
---|
1470 | static proc divisible (poly m, ideal G) |
---|
1471 | { |
---|
1472 | for (int i=1; i<=size(G); i++) |
---|
1473 | { |
---|
1474 | if (m/leadmonom(G[i])!=0) {return(1);} |
---|
1475 | } |
---|
1476 | return(0); |
---|
1477 | } |
---|
1478 | |
---|
1479 | /////////////////////////////////////////////////////////////////////////////// |
---|
1480 | |
---|
1481 | proc vanishId (list points) |
---|
1482 | "USAGE: vanishId (points); point is a list of matrices |
---|
1483 | 'points' is a list of points for which the vanishing ideal is to be |
---|
1484 | constructed |
---|
1485 | RETURN: Vanishing ideal corresponding to the given set of points |
---|
1486 | EXAMPLE: example vanishId; shows an example |
---|
1487 | " |
---|
1488 | { |
---|
1489 | int m=size(points[1]); |
---|
1490 | int n=size(points); |
---|
1491 | |
---|
1492 | ideal G=1; |
---|
1493 | int i,k,j; |
---|
1494 | list temp; |
---|
1495 | poly h,cur; |
---|
1496 | |
---|
1497 | //------------- proceed according to Farr-Gao algorithm ---------------- |
---|
1498 | for (k=1; k<=n; k++) |
---|
1499 | { |
---|
1500 | i=find_index(G,points[k]); |
---|
1501 | cur=G[i]; |
---|
1502 | for(j=i+1; j<=size(G); j++) |
---|
1503 | { |
---|
1504 | G[j]=G[j]-ev(G[j],points[k])/ev(G[i],points[k])*G[i]; |
---|
1505 | } |
---|
1506 | G=simplify(G,2); |
---|
1507 | temp=ideal2list(G); |
---|
1508 | temp=delete(temp,i); |
---|
1509 | G=list2ideal(temp); |
---|
1510 | for (j=1; j<=m; j++) |
---|
1511 | { |
---|
1512 | if (!divisible(var(j)*leadmonom(cur),G)) |
---|
1513 | { |
---|
1514 | attrib(G,"isSB",1); |
---|
1515 | h=NF((var(j)-points[k][j,1])*cur,G); |
---|
1516 | temp=ideal2list(G); |
---|
1517 | temp=insert(temp,h); |
---|
1518 | G=list2ideal(temp); |
---|
1519 | G=sort(G)[1]; |
---|
1520 | } |
---|
1521 | } |
---|
1522 | } |
---|
1523 | attrib(G,"isSB",1); |
---|
1524 | return(G); |
---|
1525 | } |
---|
1526 | example |
---|
1527 | { |
---|
1528 | "EXAMPLE:"; echo = 2; |
---|
1529 | ring r=3,(x(1..3)),dp; |
---|
1530 | |
---|
1531 | //generate all 3-vectors over GF(3) |
---|
1532 | list points=pointsGen(3,1); |
---|
1533 | |
---|
1534 | list points2=convPoints(points); |
---|
1535 | |
---|
1536 | //grasps the first 11 points |
---|
1537 | list p=graspList(points2,1,11); |
---|
1538 | print(p); |
---|
1539 | |
---|
1540 | //construct the vanishing ideal |
---|
1541 | ideal id=vanishId(p); |
---|
1542 | print(id); |
---|
1543 | } |
---|
1544 | |
---|
1545 | /////////////////////////////////////////////////////////////////////////////// |
---|
1546 | // construct the list of all vectors of length m with elements in p^e, where p |
---|
1547 | // is theharacteristic |
---|
1548 | proc pointsGen (int m, int e) |
---|
1549 | { |
---|
1550 | if (e>1) |
---|
1551 | { |
---|
1552 | list result; |
---|
1553 | int count=1; |
---|
1554 | int i,j; |
---|
1555 | list l=ringlist(basering); |
---|
1556 | int charac=l[1][1]; |
---|
1557 | number a=par(1); |
---|
1558 | list tmp; |
---|
1559 | for (i=1; i<=charac^(e*m); i++) |
---|
1560 | { |
---|
1561 | result[i]=tmp; |
---|
1562 | } |
---|
1563 | if (m==1) |
---|
1564 | { |
---|
1565 | result[count][m]=0; |
---|
1566 | count++; |
---|
1567 | for (j=1; j<=charac^(e)-1; j++) |
---|
1568 | { |
---|
1569 | result[count][m]=a^j; |
---|
1570 | count++; |
---|
1571 | } |
---|
1572 | return(result); |
---|
1573 | } |
---|
1574 | list prev=pointsGen(m-1,e); |
---|
1575 | for (i=1; i<=size(prev); i++) |
---|
1576 | { |
---|
1577 | result[count]=prev[i]; |
---|
1578 | result[count][m]=0; |
---|
1579 | count++; |
---|
1580 | for (j=1; j<=charac^(e)-1; j++) |
---|
1581 | { |
---|
1582 | result[count]=prev[i]; |
---|
1583 | result[count][m]=a^j; |
---|
1584 | count++; |
---|
1585 | } |
---|
1586 | } |
---|
1587 | return(result); |
---|
1588 | } |
---|
1589 | |
---|
1590 | if (e==1) |
---|
1591 | { |
---|
1592 | list result; |
---|
1593 | int count=1; |
---|
1594 | int i,j; |
---|
1595 | list l=ringlist(basering); |
---|
1596 | int charac=l[1][1]; |
---|
1597 | list tmp; |
---|
1598 | for (i=1; i<=charac^m; i++) |
---|
1599 | { |
---|
1600 | result[i]=tmp; |
---|
1601 | } |
---|
1602 | if (m==1) |
---|
1603 | { |
---|
1604 | for (j=0; j<=charac-1; j++) |
---|
1605 | { |
---|
1606 | result[count][m]=number(j); |
---|
1607 | count++; |
---|
1608 | } |
---|
1609 | return(result); |
---|
1610 | } |
---|
1611 | list prev=pointsGen(m-1,e); |
---|
1612 | for (i=1; i<=size(prev); i++) |
---|
1613 | { |
---|
1614 | for (j=0; j<=charac-1; j++) |
---|
1615 | { |
---|
1616 | result[count]=prev[i]; |
---|
1617 | result[count][m]=number(j); |
---|
1618 | count++; |
---|
1619 | } |
---|
1620 | } |
---|
1621 | return(result); |
---|
1622 | } |
---|
1623 | |
---|
1624 | } |
---|
1625 | |
---|
1626 | /////////////////////////////////////////////////////////////////////////////// |
---|
1627 | // convert list to a column vector |
---|
1628 | static proc list2vec (list l) |
---|
1629 | { |
---|
1630 | matrix m[size(l)][1]; |
---|
1631 | for (int i=1; i<=size(l); i++) |
---|
1632 | { |
---|
1633 | m[i,1]=l[i]; |
---|
1634 | } |
---|
1635 | return(m); |
---|
1636 | } |
---|
1637 | |
---|
1638 | /////////////////////////////////////////////////////////////////////////////// |
---|
1639 | // convert all the point in the list with list2vec |
---|
1640 | proc convPoints (list points) |
---|
1641 | { |
---|
1642 | for (int i=1; i<=size(points); i++) |
---|
1643 | { |
---|
1644 | points[i]=list2vec(points[i]); |
---|
1645 | } |
---|
1646 | return(points); |
---|
1647 | } |
---|
1648 | |
---|
1649 | /////////////////////////////////////////////////////////////////////////////// |
---|
1650 | // extracts elements from l in the range m..n |
---|
1651 | proc graspList (list l, int m, int n) |
---|
1652 | { |
---|
1653 | list result; |
---|
1654 | int count=1; |
---|
1655 | for (int i=m; i<=n; i++) |
---|
1656 | { |
---|
1657 | result[count]=l[i]; |
---|
1658 | count++; |
---|
1659 | } |
---|
1660 | return(result); |
---|
1661 | } |
---|
1662 | |
---|
1663 | /////////////////////////////////////////////////////////////////////////////// |
---|
1664 | // "characteristic" polynomial |
---|
1665 | static proc xi_gen (matrix p, int e, int s) |
---|
1666 | { |
---|
1667 | poly prod=1; |
---|
1668 | list rl=ringlist(basering); |
---|
1669 | int charac=rl[1][1]; |
---|
1670 | int l; |
---|
1671 | for (l=1; l<=s; l++) |
---|
1672 | { |
---|
1673 | prod=prod*(1-(var(l)-p[l,1])^(charac^e-1)); |
---|
1674 | } |
---|
1675 | return(prod); |
---|
1676 | } |
---|
1677 | |
---|
1678 | /////////////////////////////////////////////////////////////////////////////// |
---|
1679 | // generating polynomials in Fitzgerald-Lax construction |
---|
1680 | static proc gener_funcs (matrix check, list points, int e, ideal id, int s) |
---|
1681 | { |
---|
1682 | int n=ncols(check); |
---|
1683 | if (n!=size(points)) {ERROR("Incompatible sizes of check and points");} |
---|
1684 | ideal xi; |
---|
1685 | int i,j; |
---|
1686 | for (i=1; i<=n; i++) |
---|
1687 | { |
---|
1688 | xi[i]=xi_gen(points[i],e,s); |
---|
1689 | } |
---|
1690 | ideal result; |
---|
1691 | int m=nrows(check); |
---|
1692 | poly sum; |
---|
1693 | for (i=1; i<=m; i++) |
---|
1694 | { |
---|
1695 | sum=0; |
---|
1696 | for (j=1; j<=n; j++) |
---|
1697 | { |
---|
1698 | sum=sum+check[i,j]*xi[j]; |
---|
1699 | } |
---|
1700 | result[i]=NF(sum,id); |
---|
1701 | } |
---|
1702 | return(result); |
---|
1703 | } |
---|
1704 | |
---|
1705 | /////////////////////////////////////////////////////////////////////////////// |
---|
1706 | |
---|
1707 | proc sysFL (matrix check, matrix y, int t, int e, int s) |
---|
1708 | "USAGE: sysFL (check,y,t,e,s); check,y matrix, t,e,s int |
---|
1709 | @format |
---|
1710 | - check is a parity check matrix of the code, |
---|
1711 | - y is a received word, |
---|
1712 | - t the number of errors to correct, |
---|
1713 | - e is the extension degree, |
---|
1714 | - s is the dimension of the point for the vanishing ideal |
---|
1715 | @end format |
---|
1716 | RETURN: the system of Fitzgerald-Lax for the given decoding problem |
---|
1717 | THEORY: Based on 'check' of the given linear code, the procedure constructs |
---|
1718 | the corresponding ideal constructed with a generalization of |
---|
1719 | Cooper's philosophy. For basics of the method @ref{Fitzgerald-Lax method}. |
---|
1720 | SEE ALSO: sysQE |
---|
1721 | EXAMPLE: example sysFL; shows an example |
---|
1722 | " |
---|
1723 | { |
---|
1724 | list rl=ringlist(basering); |
---|
1725 | int charac=rl[1][1]; |
---|
1726 | int n=ncols(check); |
---|
1727 | int m=nrows(check); |
---|
1728 | list points=pointsGen(s,e); |
---|
1729 | list points2=convPoints(points); |
---|
1730 | list p=graspList(points2,1,n); |
---|
1731 | ideal id=vanishId(p,e); |
---|
1732 | ideal funcs=gener_funcs(check,p,e,id,s); |
---|
1733 | |
---|
1734 | ideal result; |
---|
1735 | poly temp; |
---|
1736 | int i,j,k; |
---|
1737 | |
---|
1738 | //--------------- add vanishing realtions --------------------- |
---|
1739 | for (i=1; i<=t; i++) |
---|
1740 | { |
---|
1741 | for (j=1; j<=size(id); j++) |
---|
1742 | { |
---|
1743 | temp=id[j]; |
---|
1744 | for (k=1; k<=s; k++) |
---|
1745 | { |
---|
1746 | temp=subst(temp,var(k),x_var(i,k,s)); |
---|
1747 | } |
---|
1748 | result=result,temp; |
---|
1749 | } |
---|
1750 | } |
---|
1751 | |
---|
1752 | //--------------- add field equations -------------------- |
---|
1753 | for (i=1; i<=t; i++) |
---|
1754 | { |
---|
1755 | for (k=1; k<=s; k++) |
---|
1756 | { |
---|
1757 | result=result,x_var(i,k,s)^(charac^e)-x_var(i,k,s); |
---|
1758 | } |
---|
1759 | } |
---|
1760 | for (i=1; i<=t; i++) |
---|
1761 | { |
---|
1762 | result=result,e(i)^(charac^e-1)-1; |
---|
1763 | } |
---|
1764 | |
---|
1765 | result=simplify(result,8); |
---|
1766 | |
---|
1767 | //--------------- add check realtions -------------------- |
---|
1768 | poly sum; |
---|
1769 | matrix syn[m][1]=syndrome(check,y); |
---|
1770 | for (i=1; i<=size(funcs); i++) |
---|
1771 | { |
---|
1772 | sum=0; |
---|
1773 | for (j=1; j<=t; j++) |
---|
1774 | { |
---|
1775 | temp=funcs[i]; |
---|
1776 | for (k=1; k<=s; k++) |
---|
1777 | { |
---|
1778 | temp=subst(temp,var(k),x_var(j,k,s)); |
---|
1779 | } |
---|
1780 | sum=sum+temp*e(j); |
---|
1781 | } |
---|
1782 | result=result,sum-syn[i,1]; |
---|
1783 | } |
---|
1784 | |
---|
1785 | result=simplify(result,2); |
---|
1786 | |
---|
1787 | points=points2; |
---|
1788 | export points; |
---|
1789 | return(result); |
---|
1790 | } |
---|
1791 | example |
---|
1792 | { |
---|
1793 | "EXAMPLE:"; echo = 2; |
---|
1794 | intvec vopt = option(get); |
---|
1795 | |
---|
1796 | list l=FLpreprocess(3,1,11,2,""); |
---|
1797 | def r=l[1]; |
---|
1798 | setring r; |
---|
1799 | int s_work=l[2]; |
---|
1800 | |
---|
1801 | //the check matrix of [11,6,5] ternary code |
---|
1802 | matrix h[5][11]=1,0,0,0,0,1,1,1,-1,-1,0, |
---|
1803 | 0,1,0,0,0,1,1,-1,1,0,-1, |
---|
1804 | 0,0,1,0,0,1,-1,1,0,1,-1, |
---|
1805 | 0,0,0,1,0,1,-1,0,1,-1,1, |
---|
1806 | 0,0,0,0,1,1,0,-1,-1,1,1; |
---|
1807 | matrix g=dual_code(h); |
---|
1808 | matrix x[1][6]; |
---|
1809 | matrix y[1][11]=encode(x,g); |
---|
1810 | //disturb with 2 errors |
---|
1811 | matrix rec[1][11]=errorInsert(y,list(2,4),list(1,-1)); |
---|
1812 | |
---|
1813 | //the Fitzgerald-Lax system |
---|
1814 | ideal sys=sysFL(h,rec,2,1,s_work); |
---|
1815 | print(sys); |
---|
1816 | option(redSB); |
---|
1817 | ideal red_sys=std(sys); |
---|
1818 | red_sys; |
---|
1819 | // read the solutions from this redGB |
---|
1820 | // the points are (0,0,1) and (0,1,0) with error values 1 and -1 resp. |
---|
1821 | // use list points to find error positions; |
---|
1822 | points; |
---|
1823 | |
---|
1824 | option(set,vopt); |
---|
1825 | } |
---|
1826 | |
---|
1827 | /////////////////////////////////////////////////////////////////////////////// |
---|
1828 | // preprocessing steps for the Fitzgerald-Lax scheme |
---|
1829 | proc FLpreprocess (int p, int e, int n, int t, string minp) |
---|
1830 | { |
---|
1831 | ring r1=p,x,dp; |
---|
1832 | int s=1; |
---|
1833 | while(p^(s*e)<n) |
---|
1834 | { |
---|
1835 | s++; |
---|
1836 | } |
---|
1837 | list var_ord; |
---|
1838 | int i,j; |
---|
1839 | int count=1; |
---|
1840 | for (i=s; i>=1; i--) |
---|
1841 | { |
---|
1842 | var_ord[count]=string("x("+string(i)+")"); |
---|
1843 | count++; |
---|
1844 | } |
---|
1845 | for (i=t; i>=1; i--) |
---|
1846 | { |
---|
1847 | var_ord[count]=string("e("+string(i)+")"); |
---|
1848 | count++; |
---|
1849 | for (j=s; j>=1; j--) |
---|
1850 | { |
---|
1851 | var_ord[count]=string("x1("+string(s*(i-1)+j)+")"); |
---|
1852 | count++; |
---|
1853 | } |
---|
1854 | } |
---|
1855 | |
---|
1856 | list rl; |
---|
1857 | list tmp; |
---|
1858 | |
---|
1859 | if (e>1) |
---|
1860 | { |
---|
1861 | rl[1]=tmp; |
---|
1862 | rl[1][1]=p; |
---|
1863 | rl[1][2]=tmp; |
---|
1864 | rl[1][2][1]=string("a"); |
---|
1865 | rl[1][3]=tmp; |
---|
1866 | rl[1][3][1]=tmp; |
---|
1867 | rl[1][3][1][1]=string("lp"); |
---|
1868 | rl[1][3][1][2]=1; |
---|
1869 | rl[1][4]=ideal(0); |
---|
1870 | } else { |
---|
1871 | rl[1]=p; |
---|
1872 | } |
---|
1873 | |
---|
1874 | rl[2]=var_ord; |
---|
1875 | |
---|
1876 | rl[3]=tmp; |
---|
1877 | rl[3][1]=tmp; |
---|
1878 | rl[3][1][1]=string("lp"); |
---|
1879 | intvec v=1; |
---|
1880 | for (i=1; i<=size(var_ord)-1; i++) |
---|
1881 | { |
---|
1882 | v=v,1; |
---|
1883 | } |
---|
1884 | rl[3][1][2]=v; |
---|
1885 | rl[3][2]=tmp; |
---|
1886 | rl[3][2][1]=string("C"); |
---|
1887 | rl[3][2][2]=intvec(0); |
---|
1888 | |
---|
1889 | rl[4]=ideal(0); |
---|
1890 | |
---|
1891 | def r2=ring(rl); |
---|
1892 | setring r2; |
---|
1893 | list l=ringlist(r2); |
---|
1894 | if (e>1) |
---|
1895 | { |
---|
1896 | execute(string("poly f="+minp)); |
---|
1897 | ideal id=f; |
---|
1898 | l[1][4]=id; |
---|
1899 | } |
---|
1900 | |
---|
1901 | def r=ring(l); |
---|
1902 | setring r; |
---|
1903 | |
---|
1904 | return(list(r,s)); |
---|
1905 | } |
---|
1906 | |
---|
1907 | /////////////////////////////////////////////////////////////////////////////// |
---|
1908 | // imitating two indeces |
---|
1909 | static proc x_var (int i, int j, int s) |
---|
1910 | { |
---|
1911 | return(x1(s*(i-1)+j)); |
---|
1912 | } |
---|
1913 | |
---|
1914 | /////////////////////////////////////////////////////////////////////////////// |
---|
1915 | // random vector of length n with entries from p^e, p the characteristic |
---|
1916 | static proc randomvector(int n, int e) |
---|
1917 | { |
---|
1918 | int i; |
---|
1919 | matrix result[n][1]; |
---|
1920 | for (i=1; i<=n; i++) |
---|
1921 | { |
---|
1922 | result[i,1]=asElement(random_prime_vector(e)); |
---|
1923 | } |
---|
1924 | return(result); |
---|
1925 | } |
---|
1926 | |
---|
1927 | /////////////////////////////////////////////////////////////////////////////// |
---|
1928 | // "convert" representation of an element from the field extension from vector |
---|
1929 | //to an elelemnt |
---|
1930 | static proc asElement(list l) |
---|
1931 | { |
---|
1932 | number s; |
---|
1933 | int i; |
---|
1934 | number w=1; |
---|
1935 | if (size(l)>1) {w=par(1);} |
---|
1936 | for (i=0; i<=size(l)-1; i++) |
---|
1937 | { |
---|
1938 | s=s+w^i*l[i+1]; |
---|
1939 | } |
---|
1940 | return(s); |
---|
1941 | } |
---|
1942 | |
---|
1943 | /////////////////////////////////////////////////////////////////////////////// |
---|
1944 | // random vector of length n with entries from p, p the characteristic |
---|
1945 | static proc random_prime_vector (int n) |
---|
1946 | { |
---|
1947 | list rl=ringlist(basering); |
---|
1948 | int i, charac; |
---|
1949 | for (i=2; i<=rl[1][1]; i++) |
---|
1950 | { |
---|
1951 | if (rl[1][1] mod i ==0) |
---|
1952 | { |
---|
1953 | break; |
---|
1954 | } |
---|
1955 | } |
---|
1956 | charac=i; |
---|
1957 | |
---|
1958 | list l; |
---|
1959 | |
---|
1960 | for (i=1; i<=n; i++) |
---|
1961 | { |
---|
1962 | l=l+list(random(0,charac-1)); |
---|
1963 | } |
---|
1964 | return(l); |
---|
1965 | } |
---|
1966 | |
---|
1967 | /////////////////////////////////////////////////////////////////////////////// |
---|
1968 | |
---|
1969 | proc decodeRandomFL(int n, int redun, int p, int e, int t, int ncodes, int ntrials, string minpol) |
---|
1970 | "USAGE: decodeRandomFL(redun,p,e,n,t,ncodes,ntrials,minpol); |
---|
1971 | @format |
---|
1972 | - n is length of codes generated, |
---|
1973 | - redun = redundancy of codes generated, |
---|
1974 | - p is the characteristic, |
---|
1975 | - e is the extension degree, |
---|
1976 | - t is the number of errors to correct, |
---|
1977 | - ncodes is the number of random codes to be processed, |
---|
1978 | - ntrials is the number of received vectors per code to be corrected, |
---|
1979 | - minpol: due to some pecularities of SINGULAR one needs to provide |
---|
1980 | minimal polynomial for the extension explicitly |
---|
1981 | @end format |
---|
1982 | RETURN: nothing |
---|
1983 | EXAMPLE: example decodeRandomFL; shows an example |
---|
1984 | " |
---|
1985 | { |
---|
1986 | intvec vopt = option(get); |
---|
1987 | |
---|
1988 | list l=FLpreprocess(p,e,n,t,minpol); |
---|
1989 | |
---|
1990 | def r=l[1]; |
---|
1991 | int s_work=l[2]; |
---|
1992 | export(s_work); |
---|
1993 | setring r; |
---|
1994 | |
---|
1995 | int i,j; |
---|
1996 | matrix h, g, word, y, rec; |
---|
1997 | ideal sys, sys2, sys3; |
---|
1998 | |
---|
1999 | |
---|
2000 | option(redSB); |
---|
2001 | matrix z[1][n]; |
---|
2002 | |
---|
2003 | for (i=1; i<=ncodes; i++) |
---|
2004 | { |
---|
2005 | h=randomCheck(redun,n,e); |
---|
2006 | g=dual_code(h); |
---|
2007 | |
---|
2008 | //---------------- generate the template system ----------------------- |
---|
2009 | sys=sysFL(h,z,t,e,s_work); |
---|
2010 | |
---|
2011 | //------ modifying the template according to the received word --------- |
---|
2012 | for (j=1; j<=ntrials; j++) |
---|
2013 | { |
---|
2014 | word=randomvector(n-redun,1); |
---|
2015 | y=encode(transpose(word),g); |
---|
2016 | print("Codeword:"); |
---|
2017 | print(y); |
---|
2018 | rec=errorRand(y,t,e); |
---|
2019 | print("Received word"); |
---|
2020 | print(rec); |
---|
2021 | sys2=LF_add_synd(rec,h,sys); |
---|
2022 | sys3=std(sys2); |
---|
2023 | print("Groebner basis of the FL system:"); |
---|
2024 | print(sys3); |
---|
2025 | } |
---|
2026 | } |
---|
2027 | |
---|
2028 | option(set,vopt); |
---|
2029 | } |
---|
2030 | example |
---|
2031 | { |
---|
2032 | "EXAMPLE:"; echo = 2; |
---|
2033 | |
---|
2034 | // correcting one error for one random binary code of length 25, |
---|
2035 | // redundancy 14; 10 words are processed |
---|
2036 | decodeRandomFL(25,14,2,1,1,1,10,""); |
---|
2037 | } |
---|
2038 | |
---|
2039 | /////////////////////////////////////////////////////////////////////////////// |
---|
2040 | // add syndrome values to the template system in FL |
---|
2041 | static proc LF_add_synd (matrix rec, matrix check, ideal sys) |
---|
2042 | { |
---|
2043 | int redun=nrows(check); |
---|
2044 | ideal result=sys; |
---|
2045 | matrix s[redun][1]=syndrome(check,rec); |
---|
2046 | for (int i=size(sys)-redun+1; i<=size(sys); i++) |
---|
2047 | { |
---|
2048 | result[i]=result[i]-s[i-size(sys)+redun,1]; |
---|
2049 | } |
---|
2050 | return(result); |
---|
2051 | } |
---|
2052 | |
---|
2053 | |
---|
2054 | /* |
---|
2055 | ////////////// SOME RELATIVELY EASY EXAMPLES ////////////// |
---|
2056 | /////////////////// THAT RUN AROUND ONE MINUTE //////////////// |
---|
2057 | |
---|
2058 | "EXAMPLE:"; echo = 2; |
---|
2059 | int q=128; int n=120; int redun=n-30; |
---|
2060 | ring r=(q,a),x,dp; |
---|
2061 | decodeRandom(n,redun,1,1,6); |
---|
2062 | |
---|
2063 | int q=128; int n=120; int redun=n-20; |
---|
2064 | ring r=(q,a),x,dp; |
---|
2065 | decodeRandom(n,redun,1,1,9); |
---|
2066 | |
---|
2067 | int q=128; int n=120; int redun=n-10; |
---|
2068 | ring r=(q,a),x,dp; |
---|
2069 | decodeRandom(n,redun,1,1,19); |
---|
2070 | |
---|
2071 | int q=256; int n=150; int redun=n-10; |
---|
2072 | ring r=(q,a),x,dp; |
---|
2073 | decodeRandom(n,redun,1,1,22); |
---|
2074 | |
---|
2075 | ////////////// SOME HARD EXAMPLES ////////////////////// |
---|
2076 | ////// THAT MAYBE WILL BE DOABLE LATER /////////////// |
---|
2077 | |
---|
2078 | 1.) These random instances are not doable in <=1000 sec. |
---|
2079 | |
---|
2080 | "EXAMPLE:"; echo = 2; |
---|
2081 | int q=128; int n=120; int redun=n-40; |
---|
2082 | ring r=(q,a),x,dp; |
---|
2083 | decodeRandom(n,redun,1,1,6); |
---|
2084 | |
---|
2085 | redun=n-30; |
---|
2086 | decodeRandom(n,redun,1,1,8); |
---|
2087 | |
---|
2088 | redun=n-20; |
---|
2089 | decodeRandom(n,redun,1,1,12); |
---|
2090 | |
---|
2091 | redun=n-10; |
---|
2092 | decodeRandom(n,redun,1,1,24); |
---|
2093 | |
---|
2094 | int q=256; int n=150; int redun=n-10; |
---|
2095 | ring r=(q,a),x,dp; |
---|
2096 | decodeRandom(n,redun,1,1,26); |
---|
2097 | |
---|
2098 | |
---|
2099 | 2.) Generic decoding is hard! |
---|
2100 | |
---|
2101 | int q=32; int n=31; int redun=n-16; int t=3; |
---|
2102 | ring r=(q,a),(V(1..n),U(n..1),s(redun..1)),(dp(n),lp(n),dp(redun)); |
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2103 | matrix check[redun][n]= 1,1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0, |
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2104 | 0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,1, |
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2105 | 0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0, |
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2106 | 0,0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0, |
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2107 | 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0, |
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2108 | 0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0, |
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2109 | 0,0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1, |
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2110 | 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1, |
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2111 | 0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0, |
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2112 | 0,0,0,0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0, |
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2113 | 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0, |
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2114 | 1,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0, |
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2115 | 0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,0, |
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2116 | 1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1, |
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2117 | 1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,0, |
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2118 | 0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,1,0,1, |
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2119 | 0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, |
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2120 | 0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0, |
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2121 | 0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,1, |
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2122 | 0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0, |
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2123 | 0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0, |
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2124 | 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0, |
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2125 | 0,1,0,1,0,0,1,0,0,1; |
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2126 | matrix rec[1][n]; |
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2127 | |
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2128 | def A=sysQE(check,rec,t,1,2); |
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2129 | setring A; |
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2130 | print(qe); |
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2131 | ideal red_qe=stdfglm(qe); |
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2132 | |
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2133 | */ |
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