Dualities of dihedral and generalised quaternion codes and applications to quantum codes

TL;DR

This paper provides a detailed algebraic analysis of dihedral and quaternion group codes, describing their duals and applying these results to construct and identify optimal quantum error-correcting codes.

Contribution

It offers a complete algebraic description of hermitian and euclidean duals of dihedral and quaternion group codes, enabling systematic quantum code construction.

Findings

Explicit dual code descriptions for dihedral and quaternion group codes

Identification of hermitian self-orthogonal dihedral codes

Construction of known optimal quantum codes using group algebra structures

Abstract

Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a complete algebraic description for the hermitian dual code of any $D_n$-code over $\mathbb{F}_{q^2}$, where $D_n$ is a dihedral group of order $2n$ with $n$ not divisible by char$(\mathbb{F}_{q^2})$, through a suitable Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_{q^2}[D_n]$, and we determine all distinct hermitian self-orthogonal $D_n$-codes over $\mathbb{F}_{q^2}$. We also present a thorough representation of the euclidean dual code of any $Q_n$-code over $\mathbb{F}_q$, where $Q_n$ is a generalised quaternion group of order $4n$ not divisible by char$(\mathbb{F}_q)$, via the Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_q[Q_n]$. In particular, since the semisimple group algebras $\mathbb{F}_{q^2}[Q_n]$ and $\mathbb{F}_{q^2}[D_{2n}]$ are isomorphic, then the hermitian dual code of any $Q_n$-code has also been fully described. As application of the hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact we rebuild some already known optimal quantum codes with this methodical approach.