Duadic negacyclic codes over a finite non-chain ring and their Gray images

TL;DR

This paper investigates duadic negacyclic codes over a specific finite non-chain ring, exploring their Gray images and constructing various dual and orthogonal codes over finite fields, with illustrative examples.

Contribution

It extends the study of duadic codes to negacyclic codes over non-chain rings and analyzes their Gray images, introducing new classes of self-dual and LCD codes.

Findings

Constructed self-dual and isodual codes over finite fields.

Analyzed Gray images of duadic negacyclic codes.

Provided examples illustrating the code constructions.

Abstract

Let $f(u)$ be a polynomial of degree $m, m \geq 2,$ which splits into distinct linear factors over a finite field $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u]/\langle f(u)\rangle$ be a finite non-chain ring. In an earlier paper, we studied duadic and triadic codes over $\mathcal{R}$ and their Gray images. Here, we study duadic negacyclic codes of Type I and Type II over the ring $\mathcal{R}$, their extensions and their Gray images. As a consequence some self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over $\mathbb{F}_q$ are constructed. Some examples are also given to illustrate this.