# On Gray Images of Cyclic and Self-Orthogonal Codes over 𝔽q + u𝔽q + v𝔽q

## TL;DR

This paper explores how certain types of codes over a specific ring can be transformed into quasi-cyclic codes using a Gray map, preserving important properties like self-orthogonality.

## Contribution

The paper introduces a Gray map for codes over a non-Frobenius ring and characterizes the resulting quasi-cyclic indices and their conditions.

## Key findings

- The Gray map preserves self-orthogonality of codes over the ring Rp,u,v.
- When gcd(n,p)=1, the Gray image of a cyclic code becomes a quasi-cyclic code of length 6n with index dividing 6.
- Only quasi-cyclic indices 1, 3, and 6 are possible, with conditions derived from the code's generators.

## Abstract

Let p be a prime with p∉{2,5} and let q=pm. This paper studies cyclic and self-orthogonal linear codes of length n over the finite local non-Frobenius ring Rp,u,v=Fq+uFq+vFq, u2=v2=uv=vu=0. We define an Fq-linear Gray map πn:Rp,u,vn→Fq6n and investigate the structural properties of Gray images of cyclic codes under this map. It is shown that πn preserves self-orthogonality and, when gcd(n,p)=1, transforms any cyclic code over Rp,u,v into a quasi-cyclic code over Fq of length 6n with index dividing 6. Moreover, we completely characterize the possible quasi-cyclic indices of the Gray images, proving that only the values l∈{1,3,6} can occur, and we establish necessary and sufficient conditions for each case in terms of the generators of the associated cyclic code. Several explicit examples are provided to illustrate the theoretical results and the resulting quasi-cyclic structures.