# Singular

#### D.10.2.2 sysCRHTMindist

Procedure from library `decodegb.lib` (see decodegb_lib).

Usage:
sysCRHTMindist(n,defset,w); n,w are int, defset is list of int's
 ``` - n length of the cyclic code, - defset is a list representing the defining set, - w is a candidate for the minimum distance ```

Return:
the ring to work with the Sala's ideal for the minimum distance containing the ideal with name 'crht_md'

Theory:
Based on 'defset' of the given cyclic code, the procedure constructs the corresponding Cooper-Reed-Heleseth-Truong ideal 'crht_md'. With its help one can find minimum distance of the code in the binary case. For basics of the method Cooper philosophy.

Example:
 ```LIB "decodegb.lib"; intvec v = option(get); // binary cyclic [15,7,5] code with defining set (1,3) list defset=1,3; // defining set int n=15; // length int d=5; // candidate for the minimum distance def A=sysCRHTMindist(n,defset,d); setring A; A; // shows the ring we are working in ==> // characteristic : 2 ==> // number of vars : 5 ==> // block 1 : ordering lp ==> // : names Z(1) Z(2) Z(3) Z(4) Z(5) ==> // block 2 : ordering C print(crht_md); // the Sala's ideal for mindist ==> Z(1)+Z(2)+Z(3)+Z(4)+Z(5), ==> Z(1)^3+Z(2)^3+Z(3)^3+Z(4)^3+Z(5)^3, ==> Z(1)^15+1, ==> Z(2)^15+1, ==> Z(3)^15+1, ==> Z(4)^15+1, ==> Z(5)^15+1, ==> Z(1)^15*Z(2)+Z(1)^14*Z(2)^2+Z(1)^13*Z(2)^3+Z(1)^12*Z(2)^4+Z(1)^11*Z(2)^5+\ Z(1)^10*Z(2)^6+Z(1)^9*Z(2)^7+Z(1)^8*Z(2)^8+Z(1)^7*Z(2)^9+Z(1)^6*Z(2)^10+Z\ (1)^5*Z(2)^11+Z(1)^4*Z(2)^12+Z(1)^3*Z(2)^13+Z(1)^2*Z(2)^14+Z(1)*Z(2)^15, ==> Z(1)^15*Z(3)+Z(1)^14*Z(3)^2+Z(1)^13*Z(3)^3+Z(1)^12*Z(3)^4+Z(1)^11*Z(3)^5+\ Z(1)^10*Z(3)^6+Z(1)^9*Z(3)^7+Z(1)^8*Z(3)^8+Z(1)^7*Z(3)^9+Z(1)^6*Z(3)^10+Z\ (1)^5*Z(3)^11+Z(1)^4*Z(3)^12+Z(1)^3*Z(3)^13+Z(1)^2*Z(3)^14+Z(1)*Z(3)^15, ==> Z(1)^15*Z(4)+Z(1)^14*Z(4)^2+Z(1)^13*Z(4)^3+Z(1)^12*Z(4)^4+Z(1)^11*Z(4)^5+\ Z(1)^10*Z(4)^6+Z(1)^9*Z(4)^7+Z(1)^8*Z(4)^8+Z(1)^7*Z(4)^9+Z(1)^6*Z(4)^10+Z\ (1)^5*Z(4)^11+Z(1)^4*Z(4)^12+Z(1)^3*Z(4)^13+Z(1)^2*Z(4)^14+Z(1)*Z(4)^15, ==> Z(1)^15*Z(5)+Z(1)^14*Z(5)^2+Z(1)^13*Z(5)^3+Z(1)^12*Z(5)^4+Z(1)^11*Z(5)^5+\ Z(1)^10*Z(5)^6+Z(1)^9*Z(5)^7+Z(1)^8*Z(5)^8+Z(1)^7*Z(5)^9+Z(1)^6*Z(5)^10+Z\ (1)^5*Z(5)^11+Z(1)^4*Z(5)^12+Z(1)^3*Z(5)^13+Z(1)^2*Z(5)^14+Z(1)*Z(5)^15, ==> Z(2)^15*Z(3)+Z(2)^14*Z(3)^2+Z(2)^13*Z(3)^3+Z(2)^12*Z(3)^4+Z(2)^11*Z(3)^5+\ Z(2)^10*Z(3)^6+Z(2)^9*Z(3)^7+Z(2)^8*Z(3)^8+Z(2)^7*Z(3)^9+Z(2)^6*Z(3)^10+Z\ (2)^5*Z(3)^11+Z(2)^4*Z(3)^12+Z(2)^3*Z(3)^13+Z(2)^2*Z(3)^14+Z(2)*Z(3)^15, ==> Z(2)^15*Z(4)+Z(2)^14*Z(4)^2+Z(2)^13*Z(4)^3+Z(2)^12*Z(4)^4+Z(2)^11*Z(4)^5+\ Z(2)^10*Z(4)^6+Z(2)^9*Z(4)^7+Z(2)^8*Z(4)^8+Z(2)^7*Z(4)^9+Z(2)^6*Z(4)^10+Z\ (2)^5*Z(4)^11+Z(2)^4*Z(4)^12+Z(2)^3*Z(4)^13+Z(2)^2*Z(4)^14+Z(2)*Z(4)^15, ==> Z(2)^15*Z(5)+Z(2)^14*Z(5)^2+Z(2)^13*Z(5)^3+Z(2)^12*Z(5)^4+Z(2)^11*Z(5)^5+\ Z(2)^10*Z(5)^6+Z(2)^9*Z(5)^7+Z(2)^8*Z(5)^8+Z(2)^7*Z(5)^9+Z(2)^6*Z(5)^10+Z\ (2)^5*Z(5)^11+Z(2)^4*Z(5)^12+Z(2)^3*Z(5)^13+Z(2)^2*Z(5)^14+Z(2)*Z(5)^15, ==> Z(3)^15*Z(4)+Z(3)^14*Z(4)^2+Z(3)^13*Z(4)^3+Z(3)^12*Z(4)^4+Z(3)^11*Z(4)^5+\ Z(3)^10*Z(4)^6+Z(3)^9*Z(4)^7+Z(3)^8*Z(4)^8+Z(3)^7*Z(4)^9+Z(3)^6*Z(4)^10+Z\ (3)^5*Z(4)^11+Z(3)^4*Z(4)^12+Z(3)^3*Z(4)^13+Z(3)^2*Z(4)^14+Z(3)*Z(4)^15, ==> Z(3)^15*Z(5)+Z(3)^14*Z(5)^2+Z(3)^13*Z(5)^3+Z(3)^12*Z(5)^4+Z(3)^11*Z(5)^5+\ Z(3)^10*Z(5)^6+Z(3)^9*Z(5)^7+Z(3)^8*Z(5)^8+Z(3)^7*Z(5)^9+Z(3)^6*Z(5)^10+Z\ (3)^5*Z(5)^11+Z(3)^4*Z(5)^12+Z(3)^3*Z(5)^13+Z(3)^2*Z(5)^14+Z(3)*Z(5)^15, ==> Z(4)^15*Z(5)+Z(4)^14*Z(5)^2+Z(4)^13*Z(5)^3+Z(4)^12*Z(5)^4+Z(4)^11*Z(5)^5+\ Z(4)^10*Z(5)^6+Z(4)^9*Z(5)^7+Z(4)^8*Z(5)^8+Z(4)^7*Z(5)^9+Z(4)^6*Z(5)^10+Z\ (4)^5*Z(5)^11+Z(4)^4*Z(5)^12+Z(4)^3*Z(5)^13+Z(4)^2*Z(5)^14+Z(4)*Z(5)^15 option(redSB); ideal red_crht_md=std(crht_md); print(red_crht_md); // reduced Groebner basis ==> Z(5)^15+1, ==> Z(4)^12+Z(4)^9*Z(5)^3+Z(4)^6*Z(5)^6+Z(4)^3*Z(5)^9+Z(5)^12, ==> Z(3)^6+Z(3)^4*Z(4)*Z(5)+Z(3)^2*Z(4)^2*Z(5)^2+Z(3)*Z(4)^4*Z(5)+Z(3)*Z(4)*Z\ (5)^4+Z(4)^6+Z(5)^6, ==> Z(2)^2+Z(2)*Z(3)+Z(2)*Z(4)+Z(2)*Z(5)+Z(3)^5*Z(4)^10*Z(5)^2+Z(3)^5*Z(4)^9*\ Z(5)^3+Z(3)^5*Z(4)^8*Z(5)^4+Z(3)^5*Z(4)^4*Z(5)^8+Z(3)^5*Z(4)^3*Z(5)^9+Z(3\ )^5*Z(4)^2*Z(5)^10+Z(3)^4*Z(4)^11*Z(5)^2+Z(3)^4*Z(4)^8*Z(5)^5+Z(3)^4*Z(4)\ ^5*Z(5)^8+Z(3)^4*Z(4)^2*Z(5)^11+Z(3)^3*Z(4)^10*Z(5)^4+Z(3)^3*Z(4)^9*Z(5)^\ 5+Z(3)^3*Z(4)^8*Z(5)^6+Z(3)^3*Z(4)^4*Z(5)^10+Z(3)^3*Z(4)^3*Z(5)^11+Z(3)^3\ *Z(4)^2*Z(5)^12+Z(3)^3*Z(5)^14+Z(3)^2*Z(4)^11*Z(5)^4+Z(3)^2*Z(4)^8*Z(5)^7\ +Z(3)^2*Z(4)^5*Z(5)^10+Z(3)^2*Z(4)^2*Z(5)^13+Z(3)^2*Z(4)*Z(5)^14+Z(3)^2+Z\ (3)*Z(4)^10*Z(5)^6+Z(3)*Z(4)^9*Z(5)^7+Z(3)*Z(4)^8*Z(5)^8+Z(3)*Z(4)^4*Z(5)\ ^12+Z(3)*Z(4)^3*Z(5)^13+Z(3)*Z(4)+Z(4)^11*Z(5)^6+Z(4)^8*Z(5)^9+Z(4)^5*Z(5\ )^12+Z(4)^3*Z(5)^14+Z(4)^2, ==> Z(1)+Z(2)+Z(3)+Z(4)+Z(5) option(set,v); ```