Subcodes#

The Subcode Constructor#

sub<C | L>: Code, List -> Code#

Given an \([n, k]\) linear code \(C\) over \(R\), construct the subcode of \(C\), generated by the elements specified by the list \(L\), where \(L\) is a list of one or more items of the following types:

(a): An element of \(C\);
(b): A set or sequence of elements of \(C\);
(c): A sequence of \(n\) elements of \(R\), defining an element of \(C\);
(d): A set or sequence of sequences of type (c);
(e): A subcode of \(C\);

Subcode(C, k): Code, RngIntElt -> Code#: Given an \([n, k]\) linear code \(C\) and an integer \(t\), \(1 \le t < n\), return a subcode of \(C\) of dimension \(t\).

Subcode(C, S): Code, { RngIntElt } -> Code#: Given an \([n, k]\) linear code \(C\) and a set \(S\) of integers, each of which lies in the range \([1,k]\), return the subcode of \(C\) generated by the basis elements whose positions appear in \(S\).

SubcodeBetweenCode(C1, C2, k): Code, Code, RngIntElt -> Code#: Given a linear code \(C_1\) and a subcode \(C_2\) of \(C_1\), return a subcode of \(C_1\) of dimension \(k\) containing \(C_2\).

SubcodeWordsOfWeight(C, S): Code, { RngIntElt } -> Code#: Given an \([n, k]\) linear code \(C\) and a set \(S\) of integers, each of which lies in the range \([1,n]\), return the subcode of \(C\) generated by those words of \(C\) whose weights lie in \(S\).

Example: Subcode Between Code (ex-db10dc)#

We give an example of how SubcodeBetweenCode may be used to create a code nested in between a subcode pair.

> C1 := RepetitionCode(GF(2),6);
> C1;
[6, 1, 6] Cyclic Code over GF(2)
Generator matrix:
[1 1 1 1 1 1]
> C3 := EvenWeightCode(6);
> C3;
[6, 5, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 0 1]
[0 0 0 0 1 1]
> C1 subset C3;
true
> C2 := SubcodeBetweenCode(C3, C1, 4);
> C2;
[6, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1 0]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 0 1]
> (C1 subset C2) and (C2 subset C3);
true

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Sum, Intersection, Duals and Hulls#

For the following operators, \(C\) and \(D\) are codes defined as subsets (or subspaces) of the same \(R\)-space \(V\).

C + D: Code, Code -> Code#: The (vector space) sum of the linear codes \(C\) and \(D\), where \(C\) and \(D\) are contained in the same \(K\)-space \(V\).

C meet D: Code, Code -> Code#: The intersection of the linear codes \(C\) and \(D\), where \(C\) and \(D\) are contained in the same \(K\)-space \(V\).

Dual(C): Code -> Code#: The dual \(D\) of the linear code \(C\). The dual consists of all codewords in the \(K\)-space \(V\) which are orthogonal to all codewords of \(C\).

Example: Sum Intersection (ex-377d6d)#

Verify some simple results from the sum and intersection of subcodes with known basis.

> C := EvenWeightCode(5);
> C;
[5, 4, 2] Linear Code over GF(2)
Generator matrix:
[1 0 0 0 1]
[0 1 0 0 1]
[0 0 1 0 1]
[0 0 0 1 1]
> C1 := sub< C | C.1 >;
> C2 := sub< C | C.4 >;
> C3 := sub< C | { C.1 , C.4} >;
> (C1 + C2) eq C3;
true
> (C1 meet C3) eq C1;
true

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Example: Dual RS (ex-cfd60d)#

Verify the orthogonality of codewords in the dual for a ReedSolomonCode.

> K<w> := GF(8);
> R := ReedSolomonCode(K, 3);
> R;
[7, 5, 3] BCH code (d = 3, b = 1) over GF(2^3)
Generator matrix:
[  1   0   0   0   0 w^3 w^4]
[  0   1   0   0   0   1   1]
[  0   0   1   0   0 w^3 w^5]
[  0   0   0   1   0   w w^5]
[  0   0   0   0   1   w w^4]
> D := Dual(R);
> D;
[7, 2, 6] Cyclic Code over GF(2^3)
Generator matrix:
[  1   0 w^3   1 w^3   w   w]
[  0   1 w^4   1 w^5 w^5 w^4]
> {<u,v> : u in R, v in D | InnerProduct(u,v) ne 0};
{}

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HermitianDual(C): CodeLinFld -> CodeLinFld#: The Hermitian dual \(D\) of the linear code \(C\). The Hermitian dual consists of all codewords in the \(K\)-space \(V\) which are orthogonal to all codewords of \(C\) with respect to the Hermitian inner product for quadratic extensions.

Hull(C): CodeLinFld -> CodeLinFld#: The (Euclidean) hull of the linear code \(C\) is the intersection of the code with its dual.

HermitianHull(C): CodeLinFld -> CodeLinFld#: The Hermitian hull of the linear code \(C\) is the intersection of the code with its Hermitian dual.

Membership and Equality#

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.