Boolean Predicates

For the following operators, C and D are codes defined as a subset (or subspace) of the R-space V.

u in C : ModTupRngElt, Code -> BoolElt
Return true if and only if the vector u of V belongs to the code C.
u notin C : ModTupRngElt, Code -> BoolElt
Return true if and only if the vector u of V does not belong to the code C.
C subset D : Code, Code -> BoolElt
Return true if and only if the code C is a subcode of the code D.
C notsubset D : Code, Code -> BoolElt
Return true if and only if the code C is not a subcode of the code D.
C eq D : Code, Code -> BoolElt
Return true if and only if the codes C and D are equal.
C ne D : Code, Code -> BoolElt
Return true if and only if the codes C and D are not equal.
IsCyclic(C) : Code -> BoolElt
Return true if and only if the linear code C is a cyclic code.
IsSelfDual(C) : Code -> BoolElt
Return true if and only if the linear code C is self-dual (or self-orthogonal) (i.e., C equals the dual of C).
IsSelfOrthogonal(C) : Code -> BoolElt
Return true if and only if the linear code C is self-orthogonal; that is, return whether C is contained in the dual of C.
IsProjective(C) : Code -> BoolElt
Returns true if and only if the (non-quantum) code C is projective.
IsZero(u) : ModTupRngElt -> BoolElt
Return true if and only if the codeword u is the zero vector.

Example CodeRng_SelfDualZ4 (H164E8)

We consider an [8, 7] linear code K8 over Z4 and examine some of its properties. 
> Z4 := IntegerRing(4);
> K8 := LinearCode< Z4, 8 |
>     [1,1,1,1,1,1,1,1],
>     [0,2,0,0,0,0,0,2],
>     [0,0,2,0,0,0,0,2],
>     [0,0,0,2,0,0,0,2],
>     [0,0,0,0,2,0,0,2],
>     [0,0,0,0,0,2,0,2],
>     [0,0,0,0,0,0,2,2]>;
> K8;
[8, 7, 2] Linear Code over IntegerRing(4)
Generator matrix:
[1 1 1 1 1 1 1 1]
[0 2 0 0 0 0 0 2]
[0 0 2 0 0 0 0 2]
[0 0 0 2 0 0 0 2]
[0 0 0 0 2 0 0 2]
[0 0 0 0 0 2 0 2]
[0 0 0 0 0 0 2 2]
> IsCyclic(K8);
true
> IsSelfDual(K8);
true
> K8 eq Dual(K8);
true
V2.28, 13 July 2023