Structure and Performance of Generalized Quasi-Cyclic Codes

TL;DR

This paper explores the structure and properties of generalized quasi-cyclic (GQC) codes, providing new theoretical insights, bounds, and explicit examples of LCD GQC codes that extend beyond traditional QC codes.

Contribution

It introduces a decomposition of GQC codes via ring theory, derives new bounds and criteria, and constructs explicit LCD GQC codes that are not quasi-cyclic.

Findings

Decomposition of GQC codes into concatenated codes.

Trace formula and minimum distance bounds for GQC codes.

Explicit examples of LCD GQC codes that are not QC.

Abstract

Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.