Generalized Quasi-Cyclic Codes Over $\mathbb{F}_q+u\mathbb{F}_q$

TL;DR

This paper explores the structure, decomposition, and properties of generalized quasi-cyclic codes over the ring f_q + uf_q, providing new insights into their minimal generators and minimum distance bounds.

Contribution

It introduces a comprehensive analysis of GQC codes over f_q + uf_q, including structural properties, decomposition via the Chinese Remainder Theorem, and bounds for 1-generator codes.

Findings

Structural properties and decomposition of GQC codes are established.

Minimal generating sets for 1-generator GQC codes are derived.

Lower bounds on the minimum distance of GQC codes are provided.

Abstract

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $u^2=0$, $q=p^n$, $n$ a positive integer and $p$ a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimal generating sets and lower bounds on the minimum distance are given. As a special class of GQC codes, quasi-cyclic (QC) codes over $\mathbb{F}_q+u\mathbb{F}_q$ are also discussed briefly in this paper.