Canonization of linear codes over $\mathbb Z$_{4}

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

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

Abstract:
Two linear codes $C, C' \leq \mathbb Z$_{4}^{n} 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.

References