On the Structure of the Linear Codes with a Given Automorphism

TL;DR

This paper investigates the structure of linear codes over finite fields that possess a permutation automorphism of a certain order, focusing on their relation to quasi-cyclic codes and conditions for various duality properties.

Contribution

It provides a detailed structural analysis of such codes and establishes necessary and sufficient conditions for self-orthogonality, self-duality, and linear complementary duality.

Findings

Characterization of codes with permutation automorphisms

Conditions for self-orthogonality and self-duality

Relation to quasi-cyclic and almost quasi-cyclic codes

Abstract

The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual.