$(1-2u^3)$-constacyclic codes and quadratic residue codes over $\mathbb{F}_{p}[u]/\langle u^4-u\rangle$

TL;DR

This paper investigates a special class of constacyclic and quadratic residue codes over a specific finite ring, exploring their properties, equivalences, and Gray images, with examples for prime powers and applications to self-dual codes.

Contribution

It introduces and analyzes $(1-2u^3)$-constacyclic codes over a non-chain ring, establishes their equivalence to cyclic codes, and constructs self-dual codes via Gray maps.

Findings

Explicit examples of codes for specific primes and lengths.

Gray map preserves self-duality and yields self-dual codes over finite fields.

Extension of quadratic residue codes over the ring $ $.

Abstract

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^4$ is defined which preserves self duality and gives self-dual and formally self-dual codes over $\mathbb{F}_{p}$ from extended quadratic residue codes.