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A length n additive code has an alphabet F and
coefficient field K ⊆F.
The code consists of codewords which are a K-linear subspace of F(n).
Field(C) : Code -> Rng
The underlying field (or alphabet) of the codewords of the additive code C.
A length n additive code with alphabet F consists of codewords from
F(n).
The field over which the codewords of the additive code C are
considered linear.
This will be a subfield of the alphabet of C.
The ambient space of the additive code C,
i.e. the generic R-space V in which C is contained.
Given a length n additive code C,
return the generic [n, n, 1] code in which C is contained.
A code can often be represented using several different coefficient fields.
> F<w> := GF(5,4);
> K := GF(5,2);
> C := RandomAdditiveCode(F, K, 12, 5);
> C:Minimal;
[12, 2 1/2 : 5] GF(5^2)-Additive Code over GF(5^4)
> #C;
9765625
> Alphabet(C);
Finite field of size 5^4
> CoefficientField(C);
Finite field of size 5^2
>
> C1 := AdditiveCode(GF(5), C);
> C1:Minimal;
[12, 2 1/2 : 10] GF(5)-Additive Code over GF(5^4)
> #C1;
9765625
> Alphabet(C1);
Finite field of size 5^4
> CoefficientField(C1);
Finite field of size 5
Return the block length n of an additive code C.
The (rational) dimension k of C. If the alphabet of C is F, then
the dimension is defined by the equation #C = (#F)k.
Note that since any basis of the additive code C is
relative to the coefficient field K,
this dimension is not necessarily equal to the number of generators of C
and is not even necessarily integral.
Ngens(C) : Code -> RngIntElt
The number of generators of the additive code C. Note that if the coefficient
ring of C is not the same as its alphabet then this will be different from
the dimension of C.
Given an additive code C, return the number of codewords belonging to C.
The information rate of the [n, k] code C. This is the ratio k/n.
BasisMatrix(C) : Code -> ModMatRngElt
The generator matrix for an [n, k(kg)] K-additive code C over F is a
kg x n matrix over F, whose kg rows form a basis for C when
considered as vectors over K.
Generators(C) : Code -> { ModTupFldElt }
A basis for the K-additive code C,
returned as a sequence of codewords over the
alphabet of C, which generate the code over K.
Given an [n, k(kg)] K-additive code C and a
positive integer i, 1 ≤i ≤kg,
return the i-th element of the current basis of C over K.
The code that is dual to the code C. For an additive code C, this
is the nullspace with respect to the trace inner product of the
coefficient field.
The parity check matrix for the code C, returned as
an element of Hom(V, U).
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