Self-Dual Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and Their Applications in the Study of Quasi-Abelian Codes

TL;DR

This paper characterizes and enumerates Hermitian self-dual codes over a specific finite ring and explores their applications in classifying and constructing quasi-abelian codes in group algebras, especially over fields of characteristic 3.

Contribution

It provides a complete characterization and enumeration of Hermitian self-dual codes over the ring and applies these results to study and explicitly construct self-dual quasi-abelian codes in group algebras.

Findings

Complete classification of Hermitian self-dual codes over the ring

Explicit enumeration formulas for self-dual quasi-abelian codes in characteristic 3

Construction methods for self-dual codes over the specified ring

Abstract

Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring~$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ with $u^3=0$ have been established. In this paper, Hermitian self-dual linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ are studied for all square prime powers~$q$. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ is studied, where $H\leq G$ are finite abelian groups and $\mathbb{F}_q[H]$ is a principal ideal group algebra. General characterization and enumeration of $H$-quasi-abelian codes and self-dual $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ are given. For the special case where the field characteristic is $3$, an explicit formula for the number of self-dual $A\times \mathbb{Z}_3$-quasi-abelian codes in $\mathbb{F}_{3^m}[A\times \mathbb{Z}_3\times B]$ is determined for all finite abelian groups $A$ and $B$ such that $3\nmid |A|$ as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over $\mathbb{F}_{3^m}+u\mathbb{F}_{3^m}+u^2\mathbb{F}_{3^m}$.