Some MDS codes over dihedral groups
Yuchao Wang
TL;DR
This work addresses constructing maximum-distance-separable (MDS) codes within the non-Abelian setting of dihedral group algebras $F_qD_{2n}$ by exploiting the Wedderburn decomposition under the assumption that $\mathrm{char}(F_q)\nmid |D_{2n}|$ and $F_q$ is the splitting field for $D_{2n}$. The authors derive the explicit decomposition into simple components and central primitive idempotents, then realize left ideals formed from these idempotents as dihedral group codes. They prove the existence of $[2n,2n-2,3]$ and $[2n,2n-3,4]$ MDS dihedral codes for odd $n$ under the condition $2n|q-1$ (with refinements for various choices of idempotents and scalars), and illustrate with a concrete $F_{5^2}$ example showing $[6,4,3]$ and $[6,3,4]$ codes. The results extend the catalog of MDS codes in non-Abelian group algebras and provide constructive methods via explicit idempotents and Wedderburn blocks, with potential applications in coding theory where dihedral symmetry is relevant.
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.