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D.10.1.8 prepSV

Procedure from library brnoeth.lib (see brnoeth_lib).

prepSV( G, D, F, EC ); G,D,F intvecs and EC a list

list E of size n+3, where n=size(D). All its entries but E[n+3] are matrices:
   E[1]:  parity check matrix for the current AG code
   E[2] ... E[n+2]:  matrices used in the procedure decodeSV
   E[n+3]:  intvec with
       E[n+3][1]: correction capacity 
 of the algorithm
       E[n+3][2]: designed Goppa distance 
 of the current AG code

Computes the preprocessing for the basic (Skorobogatov-Vladut) decoding algorithm.
The procedure must be called within the ring EC[1][4], where EC is the output of extcurve(d) (or in the ring EC[1][2] if d=1)
The intvec G and F represent rational divisors (see BrillNoether for more details).
The intvec D refers to rational places (see AGcode_Omega for more details.). The current AG code is AGcode_Omega(G,D,EC).
If you know the exact minimum distance d and you want to use it in decodeSV instead of $delta$, you can change the value of E[n+3][2] to d before applying decodeSV.
If you have a systematic encoding for the current code and want to keep it during the decoding, you must previously permute D (using permute_L(D,P);), e.g., according to the permutation P=L[3], L being the output of sys_code.

F must be a divisor with support disjoint from the support of D and with degree $epsilon + genus$, where

G should satisfy $ 2*genus-2 < deg(G) < size(D) $, which is not checked by the algorithm.
G and D should also have disjoint supports (checked by the algorithm).

LIB "brnoeth.lib";
int plevel=printlevel;
ring s=2,(x,y),lp;
list HC=Adj_div(x3+y2+y);
==> The genus of the curve is 1
==> Total number of rational places : NrRatPl = 9
def ER=HC[1][4];
setring ER;
intvec G=5;      // the rational divisor G = 5*HC[3][1]
intvec D=2..9;   // D = sum of the rational places no. 2..9 over F_4
// construct the corresp. residual AG code of type [8,3,>=5] over F_4:
matrix C=AGcode_Omega(G,D,HC);
==> Vector basis successfully computed 
// we can correct 1 error and the genus is 1, thus F must have degree 2
// and support disjoint from that of D;
intvec F=2;
list SV=prepSV(G,D,F,HC);
==> Vector basis successfully computed 
==> Vector basis successfully computed 
==> Vector basis successfully computed 
// now everything is prepared to decode with the basic algorithm;
// for example, here is a parity check matrix to compute the syndrome :
==> 0,0,1,  1,    (a+1),(a),  (a),  (a+1),
==> 0,1,(a),(a+1),(a),  (a+1),(a),  (a+1),
==> 1,1,1,  1,    1,    1,    1,    1,    
==> 0,0,1,  1,    (a),  (a+1),(a+1),(a),  
==> 0,0,(a),(a+1),1,    1,    (a+1),(a)   
// and here you have the correction capacity of the algorithm :
int epsilon=SV[size(D)+3][1];
==> 1
See also: AGcode_Omega; decodeSV; extcurve; permute_L; sys_code.