New Codes From Old

Combining two or more codes

u_u_plus_v and u_plus_w_v_plus_w_u_plus_v_plus_w throw an error when one of the arguments is the zero code. For the latter, let C1 be an $[n, k_1, d_1]$ and C2 an $[n, k_2, d_2]$ linear code; the construction produces an $[3n, 2k_1 + k_2]$ linear code, and for binary codes

\[\mathrm{wt}(u + w \mid v + w \mid u + v + w) = 2\,\mathrm{wt}(u \veebar v) - \mathrm{wt}(w) + 4s, \qquad s = |\{i \mid u_i = v_i = 0,\, w_i = 1\}|.\]

Construction X takes an $[n, k, d]$ code C1, an $[n, k - l, d + e]$ code C2, and an $[m, l, e]$ code C3 with C2 a proper subcode of C1, and returns an $[n + m, k, d + e]$ code. Construction X3 takes $[n, k_1, d_1]$, $[n, k_2, d_2]$, $[n, k_3, d_3]$, $[n_4, k_2 - k_1, d_4]$, and $[n_5, k_3 - k_2, d_5]$ codes with C1 ⊂ C2 ⊂ C3 and returns an $[n + n_4 + n_5, k_3, d]$ code with $d \geq \min\{d_1, d_2 + d_4, d_3 + d_5\}$.

The direct sum ⊕ has generator matrix G1 ⊕ G2 and parity-check matrix H1 ⊕ H2. The generator matrix of the direct product × is the Kronecker product of the input generator matrices, and the parity-check matrix of the tensor product is the Kronecker product of the input parity-check matrices. There is some debate over how to define the entrywise (Schur) product of two codes; the result is known to often be the full ambient space.

juxtaposition is representation dependent and therefore works on the potentially overcomplete generator matrices rather than on the standard form.

Modifying a single code

Extending adds a column to the generator matrix whose values make the row sums zero. This even extension is the default for extend(C), and the new column may instead be inserted at any index with extend(C, c). In the general case one supplies a vector a and the new entries are the negated inner products of a with the rows; the standard definition is the special case where a is the all-ones vector.

Puncturing deletes columns from the generator matrix and removes any resulting zero rows. Expurgating deletes rows and removes any resulting zero columns, working directly on the potentially overcomplete generator matrix rather than the standard form. Shortening is expurgating followed by puncturing; the implementation uses the theorem that the code shortened on L is the dual of the dual punctured on L, that is dual(puncture(dual(C), L)).

Augmentation vertically joins a matrix to the bottom of the generator matrix, again working on the potentially overcomplete form. Lengthening augments the all-ones row and then extends.

subcode_of_dimension_between_codes adds generators of C1 / C2 to C2 until the desired dimension is reached. The subfield subcode is computed directly via an expansion, whereas the trace code is computed using Delsarte's theorem.

If C is a quasi-cyclic code, permute_code returns a LinearCode.

Experimental

even_subcode, doubly_even_subcode, and triply_even_subcode need significantly more testing, but appear to work so far.

Base.:/ — Method

/(
    C2::AbstractLinearCode,
    C1::AbstractLinearCode
) -> LinearCode

Return the complementary subcode C2 / C1 given C1 ⊆ C2.

New Codes From Old

Combining two or more codes

Modifying a single code

Weight reduction