The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes

Pages: 363 - 383, Volume 3, Issue 4, November 2009

doi:10.3934/amc.2009.3.363       Abstract        Full Text (283.1K)       Related Articles

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

Abstract: The main aim of the classification of linear codes is the evaluation of complete lists of representatives of the isometry classes. These classes are mostly defined with respect to linear isometry, but it is well known that there is also the more general definition of semilinear isometry taking the field automorphisms into account. This notion leads to bigger classes so the data becomes smaller. Hence we describe an algorithm that gives canonical representatives of these bigger classes by calculating a unique generator matrix to a given linear code, in a well defined manner.
    The algorithm is based on the partitioning and refinement idea which is also used to calculate the canonical labeling of a graph [12] and it similarly returns the automorphism group of the given linear code. The time needed by the implementation of the algorithm is comparable to Leon's program [10] for the calculation of the linear automorphism group of a linear code, but it additionally provides a unique representative and the automorphism group with respect to the more general notion of semilinear equivalence. The program can be used online under http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/CanonicalForm/index.html.

Keywords:  Automorphism group, canonization, coding theory, error-correcting code, group action, representative, semilinear isometry.
Mathematics Subject Classification:  Primary: 05E20; Secondary: 20B25, 94B05.

Received: May 2009;      Revised: September 2009;      Published: November 2009.