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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
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- 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
==> // coefficients: ZZ/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);
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