Advances in Mathematics of Communications (AMC)

Canonization of linear codes over $\mathbb Z$4

Pages: 245 - 266, Volume 5, Issue 2, May 2011      doi:10.3934/amc.2011.5.245

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Thomas Feulner - Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany (email)

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.

Keywords:  Automorphism group, canonization, coding theory, group action, representative, isometry, $\mathbb Z$4-linear code.
Mathematics Subject Classification:  Primary: 05E20; Secondary: 20B25, 94B05.

Received: April 2010;      Revised: October 2010;      Available Online: May 2011.