On the Structure of the Linear Codes with a Given Automorphism
Stefka Bouyuklieva
TL;DR
The paper addresses the structure of linear codes over $\mathbb{F}_q$ with a permutation automorphism of order $m$, showing they form generalized quasi-cyclic (GQC) structures and establishing conditions for self-orthogonality, self-duality, and LCD. It develops a decomposition framework using $F_{\sigma}(C)$, $E_{\sigma}(C)$, and projection maps $\pi$ and $\psi$, with distinct treatments for cases where $\gcd(m,\mathrm{char}(\mathbb{F}_q))=1$ and where $m$ shares factors with the characteristic. For coprime cases, $C=F_{\sigma}(C) \oplus E_{\sigma}(C)$ and the projection $C_{\pi}$ preserves duality properties; in QC and almost QC settings, an explicit ring decomposition of $\mathcal{R}_m$ and corresponding constituent codes describe when $C$ is self-dual, self-orthogonal, or LCD. When $m$ is not coprime to the characteristic, the paper leverages $\psi$ and $\pi$ to relate $C$, $C_{\psi}$, and $C_{\pi}$, with $C_{\pi} \subseteq C_{\psi}^{\perp}$ and equality in the self-dual case under certain conditions. The framework yields practical code constructions and several explicit examples, demonstrating the approach’s effectiveness for generating optimal codes with prescribed automorphisms and properties.
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.