Canonization of linear codes over $\mathbb Z$4

Thomas Feulner

doi:10.3934/amc.2011.5.245

Article Contents

2011, Volume 5, Issue 2: 245-266. Doi: 10.3934/amc.2011.5.245

This issue Previous Article Algebraic structure of the minimal support codewords set of some linear codes Next Article On the performance of binary extremal self-dual codes

Canonization of linear codes over $\mathbb Z$₄

Thomas Feulner^1,

1.
Department of Mathematics, University of Bayreuth, 95440 Bayreuth

Received: April 2010

Revised: October 2010

Published online: May 2011

Abstract / Introduction Related Papers Cited by

Abstract

Two linear codes $C, C' \leq \mathbb Z$₄ⁿ are equivalent if there is a permutation $\pi \in S_n$ of the coordinates and a vector $\varphi \in \{1,3\}^n$ of column multiplications such that $(\varphi; \pi) C = C'$. This generalizes the notion of code equivalence of linear codes over finite fields.
In a previous paper, the author has described an algorithm to compute the canonical form of a linear code over a finite field. In the present paper, an algorithm is presented to compute the canonical form as well as the automorphism group of a linear code over $\mathbb Z$₄. This solves the isomorphism problem for $\mathbb Z$₄-linear codes. An efficient implementation of this algorithm is described and some results on the classification of linear codes over $\mathbb Z$₄ for small parameters are discussed.

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Mathematics Subject Classification: Primary: 05E20; Secondary: 20B25, 94B05.

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