Five infinite families of binary cyclic codes and their related codes with good parameters

TL;DR

This paper constructs and analyzes five infinite families of binary cyclic codes with good parameters, including distance-optimal and self-dual codes, enhancing coding theory applications.

Contribution

It introduces five new infinite families of binary cyclic codes with improved parameters and analyzes their properties, including duals and self-dual codes.

Findings

Three families contain distance-optimal codes.

Two families are composed of duadic codes with square-root-like bounds.

Two infinite families of self-dual codes with good bounds.

Abstract

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters $[n, k]$ and $(n-6)/3 \leq k \leq 2(n+6)/3$. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained.