Two classes of narrow-sense BCH codes and their duals

TL;DR

This paper investigates the duals of narrow-sense BCH codes, providing conditions for when they are dually-BCH, establishing lower bounds on their minimum distances, and solving related conjectures and open problems.

Contribution

It extends the understanding of duals of BCH codes by giving necessary and sufficient conditions, and introduces bounds and solutions to existing conjectures.

Findings

Conditions for BCH codes to be dually-BCH established.

Lower bounds on dual code minimum distances derived.

Proved a conjecture on the largest coset leader modulo n.

Abstract

BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of the duals of BCH codes. Recently, a new concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in \cite{GDL21}. For a prime power $q$ and an integer $m \ge 4$, let $n=\frac{q^m-1}{q+1}$ \ ($m$ even), or $n=\frac{q^m-1}{q-1}$ \ ($q>2$). In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense BCH codes of length $n$ are dually-BCH codes, which extended the results in \cite{GDL21}. Lower bounds on the minimum distances of their dual codes are developed for $n=\frac{q^m-1}{q+1}$ \ ($m$ even). As byproducts, we present the largest coset leader $\delta_1$ modulo $n$ being of two types, which proves a conjecture in \cite{WLP19} and partially solves an open problem in \cite{Li2017}. We also investigate the parameters of the narrow-sense BCH codes of length $n$ with design distance $\delta_1$. The BCH codes presented in this paper have good parameters in general.