Reversible cyclic codes over $\mathbb{F}_q + u \mathbb{F}_q$

TL;DR

This paper investigates the structure and classification of reversible cyclic codes over the ring f_q + uf_q, providing generator sets, dual code properties, and illustrative examples to advance coding theory over rings.

Contribution

It introduces a unique set of generators for cyclic codes over f_q + uf_q and classifies reversible cyclic codes, including dual code reversibility conditions.

Findings

Derived a unique generator set for cyclic codes over the ring

Classified reversible cyclic codes based on generators

Showed duals of reversible codes are reversible under certain conditions

Abstract

Let $q$ be a power of a prime $p$. In this paper, we study reversible cyclic codes of arbitrary length over the ring $ R = \mathbb{F}_q + u \mathbb{F}_q$, where $u^2=0 mod q$. First, we find a unique set of generators for cyclic codes over $R$, followed by a classification of reversible cyclic codes with respect to their generators. Also, under certain conditions, it is shown that dual of reversible cyclic code is reversible over $\mathbb{Z}_2+u\mathbb{Z}_2$. Further, to show the importance of these results, some examples of reversible cyclic codes are provided.