One generator quasi-cyclic codes over F2 + uF2 + vF2 + uvF2

TL;DR

This paper investigates the structure and properties of 1-generator quasi-cyclic and generalized quasi-cyclic codes over a specific ring, providing minimal spanning sets and a BCH type bound, advancing coding theory over rings.

Contribution

It introduces the structure of 1-generator quasi-cyclic codes over a ring with nilpotent elements and extends to generalized codes with a BCH bound, a novel generalization.

Findings

Determined minimal spanning sets for these codes.

Established a BCH type bound for generalized quasi-cyclic codes.

Extended the understanding of code structures over rings with nilpotent elements.

Abstract

In this paper, we study the structure of 1-generator quasi-cyclic codes over the ring R = F2 + uF2 + vF2 + uvF2, with u2 = v2 = 0 and uv = vu. We determine the minimal spanning sets for these codes. As a generalization of these codes, we also investigate the structure of 1-generator generalized quasi-cyclic codes over R and determine a BCH type bound for them.