Program that calculates the canonical generator matrix of a linear code and the automorphism group under the action of the linear or semilinear isometry group (article)

List of pairwise CCZ-inequivalent APN Functions

This program computes a unique generator matrix under all generator matrices of (semi-)linearly isometric, linear codes over arbitrary fields, the field elements are numbered by 0, ..., q-1 with q = p^r, p prime. The numbering N is based on a root ξ of the Conway Polynomial of degree r:

μ = ∑_{i = 0,..., r-1} μ_i ξⁱ ⇒ N(μ) = ∑_{i = 0,..., r-1} μ_i pⁱ

You can use any generator matrix over the fields GF(q) with q = 2, 3, 4, 5, 7, 8, 9, 16, 32.

The program returns the elements of the acting group by tupels (A, φ, α, π) with an invertible k × k matrix A, a vector of nonzero column multiplications φ, a field automorphism α and a permutation of columns π .

Please, enter some generator matrix in the following form.

Example: The quadratic residue code C(11,6) over GF(3). n=11, k=6, q=3:

2 2 1 2 0 1 0 0 0 0 0
0 2 2 1 2 0 1 0 0 0 0
0 0 2 2 1 2 0 1 0 0 0
0 0 0 2 2 1 2 0 1 0 0
0 0 0 0 2 2 1 2 0 1 0
0 0 0 0 0 2 2 1 2 0 1

You can also use this code for the field GF(9), where you can see the difference between semilinear and linear isometry.

The algorithm is implemented in the programming language C++, but we use the computer algebra system Magma to compute the group of known automorphism in the backtrack search.

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semilinear
linear
n:
k:
q:
generator matrix:

The execution time of this online version is limited to 2 minutes.

Last Modified: 2012-07-06.