New Binary and Ternary Quasi-Cyclic Codes with Good Properties

TL;DR

This paper introduces 113 new binary quasi-cyclic codes with optimal parameters and desirable properties, along with an algorithm to generate additional codes, significantly advancing coding theory.

Contribution

The paper presents 113 new binary QC codes with optimal parameters and properties, and introduces a novel algorithm for generating more codes, including 35 record-breaking codes.

Findings

113 new binary QC codes found with optimal parameters

Many codes are reversible, LCD, self-orthogonal, or dual-containing

35 new record-breaking linear codes generated

Abstract

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 35 new record breaking linear codes produced from this method.