The Permutation Groups and the Equivalence of Cyclic and Quasi-Cyclic Codes

TL;DR

This paper characterizes the permutation groups associated with cyclic and quasi-cyclic codes over finite fields, extending previous results and identifying permutation sets that establish code equivalence.

Contribution

It identifies the classes of permutation groups for cyclic codes and extends the analysis to quasi-cyclic codes, providing new insights into code equivalence.

Findings

Characterization of permutation groups for cyclic codes

Extension of results to quasi-cyclic codes

Identification of permutations establishing code equivalence

Abstract

We give the class of finite groups which arise as the permutation groups of cyclic codes over finite fields. Furthermore, we extend the results of Brand and Huffman et al. and we find the properties of the set of permutations by which two cyclic codes of length p^r can be equivalent. We also find the set of permutations by which two quasi-cyclic codes can be equivalent.