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: Self Dual Z4 (ex-659440)

We consider an [8, 7] linear code \(K_8\) over \(Z_4\) 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

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