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

Miguel Sales-Cabrera, Xaro Soler-Escrivà, Víctor Sotomayor

TL;DR

The work provides a complete algebraic framework for dualities of dihedral and generalized quaternion group codes via precise Wedderburn–Artin decompositions, enabling explicit description of hermitian duals for $D_n$-codes over $\mathbb{F}_{q^2}$ and Euclidean duals for $Q_n$-codes over $\mathbb{F}_q$. By refining the decomposition and carefully tracking blockwise duals, the authors derive necessary and sufficient conditions for hermitian self-orthogonality and enumerate such codes, which in turn underpins systematic constructions of CSS quantum codes. The paper also demonstrates how these dualities yield known optimal quantum codes and provides corrections to prior results in the literature. Through detailed examples, it showcases the practical power of algebraic group-code methods to design quantum-error-correcting codes with favorable parameters.

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 $\gcd(q,n)=1$, 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 $4n$ with $\gcd(q,4n)=1$, 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.

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

TL;DR

The work provides a complete algebraic framework for dualities of dihedral and generalized quaternion group codes via precise Wedderburn–Artin decompositions, enabling explicit description of hermitian duals for -codes over and Euclidean duals for -codes over . By refining the decomposition and carefully tracking blockwise duals, the authors derive necessary and sufficient conditions for hermitian self-orthogonality and enumerate such codes, which in turn underpins systematic constructions of CSS quantum codes. The paper also demonstrates how these dualities yield known optimal quantum codes and provides corrections to prior results in the literature. Through detailed examples, it showcases the practical power of algebraic group-code methods to design quantum-error-correcting codes with favorable parameters.

Abstract

Let be a finite field of elements, for some prime power , and let be a finite group. A (left) group code, or simply a -code, is a (left) ideal of the group algebra . In this paper, we provide a complete algebraic description for the hermitian dual code of any -code over , where is a dihedral group of order with , through a suitable Wedderburn-Artin's decomposition of the group algebra , and we determine all distinct hermitian self-orthogonal -codes over . We also present a thorough representation of the euclidean dual code of any -code over , where is a generalised quaternion group of order with , via the Wedderburn-Artin's decomposition of the group algebra . In particular, since the semisimple group algebras and are isomorphic, then the hermitian dual code of any -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.

Paper Structure

This paper contains 15 sections, 23 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group codes
  4. The Wedderburn-Artin's decomposition of certain group algebras
  5. Dualities of group codes
  6. On the euclidean dual of a Dn-code
  7. CSS quantum codes
  8. On hermitian dualities of dihedral codes
  9. A suitable Wedderburn-Artin's decomposition of Fq2[Dn]
  10. On the hermitian dual of a Dn-code
  11. Computing hermitian self-orthogonal Dn-codes
  12. On euclidean dualities of generalised quaternion codes
  13. The Wedderburn-Artin's decomposition of Fq[Qn]
  14. On the euclidean dual of a Qn-code
  15. Some numerical examples

Key Result

Theorem 2.1

(Wedderburn-Artin's decomposition for finite group algebras). Let $G$ be a finite group such that $\mathbb{F}_q[G]$ is a semisimple group algebra. Then $\mathbb{F}_q[G]$ is isomorphic, as $\mathbb{F}_q$-algebra, to the direct sum of some matrix rings over suitable extensions of $\mathbb{F}_q$. Speci satisfying $|G| = \sum_{i=1}^{s} n_i^2 r_i$.

Theorems & Definitions (51)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 41 more