Some MDS codes over dihedral groups

TL;DR

This paper constructs specific maximum distance separable (MDS) codes over dihedral group algebras using Wedderburn decomposition and primitive idempotents, expanding coding theory over algebraic structures.

Contribution

It provides explicit constructions of MDS codes over dihedral groups using algebraic decompositions and primitive idempotents, under certain field conditions.

Findings

Constructed $[2n,2n-2,3]$ MDS codes over dihedral groups.

Constructed $[2n,2n-3,4]$ MDS codes over dihedral groups.

Derived Wedderburn decomposition and primitive idempotents for $F_qD_{2n}$.

Abstract

In this paper,we show some $[2n,2n-2,3]$ and $[2n,2n-3,4]$ MDS codes over dihedral codes $F_qD_{2n}$,in the case $n$ is odd and char$F_q$$\nmid$$\lvert G \rvert$ and $F_q$ contains primitive root of exponent $\lvert G \rvert$ i.e $F_q$ is the splitting field of $G$.Before that,we will give the Wedderburn decomposition and specific forms of linear primitive idempotents of $F_qD_{2n}$ under the above conditions.The MDS codes we construct are obtained by its Wedderburn decomposition and linear primitive idempotents.