Binary duadic codes and their related codes with a square-root-like lower bound

TL;DR

This paper constructs new binary duadic codes with lengths of 2^m-1, providing a square-root-like lower bound on their minimum distances, thereby addressing an open problem in the field.

Contribution

It introduces several families of binary duadic codes with specific parameters and establishes a square-root-like lower bound on their minimum distances, solving a previously open problem.

Findings

Constructed binary duadic codes with length 2^m-1 and dimension 2^(m-1)

Established a square-root-like lower bound on minimum distances

Provided parameters for dual and extended codes, including self-dual and doubly-even codes

Abstract

Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it is hard to construct infinite families of binary cyclic codes [n, n+1/2] with good minimum distance. In this paper, by using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present several families of binary duadic codes with length 2^m-1 and dimension 2^(m-1), and the minimum distances have a square-root-like lower bound. As a by-product, the parameters of their dual codes and extended codes are provided, where the latter are self-dual and doubly-even.