Some properties of dihedral group codes

TL;DR

This paper provides an explicit algebraic description of dihedral group codes, explores their duals and self-duality criteria, and connects them to cyclic code theory with practical examples.

Contribution

It introduces a detailed algebraic framework for dihedral codes, including duals, self-duality conditions, and their relation to cyclic codes, which was not previously fully characterized.

Findings

Explicit algebraic descriptions of dihedral codes and their duals.

Criteria for self-duality of dihedral codes.

Connections established between dihedral and cyclic codes.

Abstract

In this paper, we study the dihedral codes, i.e. the left ideals of $\mathbb{F}_qD_{n}$ in the case $\gcd(q, n) = 1$. An explicit algebraic description of the dihedral codes and their duals is obtained. In addition, a criterion for self-duality of a dihedral code is obtained. Bases, generating and check matrices of dihedral codes are constructed. Given a dihedral code, we consider exterior and interior codes that are induced by cyclic codes. Using this codes, some properties of generating matrices are described and a connection to cyclic code theory is established. In addition, some estimates of code parameters are obtained and several illustrative examples are given.