Three classes of BCH codes and their duals

TL;DR

This paper investigates three specific classes of BCH codes, providing precise parameters and conditions for their duals, which enhances understanding of their structure and potential applications in communication systems.

Contribution

The paper determines parameters for two classes of BCH codes and establishes conditions for duality in the third class, advancing the theoretical knowledge of BCH code structures.

Findings

Parameters of _{(q,n,,1)} and its duals are precisely determined.

Conditions for _{(q,n,,2)} to be dually-BCH are established.

Enhanced understanding of the structure and duality of three classes of BCH codes.

Abstract

BCH codes are an important class of cyclic codes, and have wide applicantions in communication and storage systems. However, it is difficult to determine the parameters of BCH codes and only a few cases are known. In this paper, we mainly study three classes of BCH codes with $n=q^{m}-1,\frac{q^{2s}-1}{q+1},\frac{q^{m}-1}{q-1}$. On one hand, we accurately give the parameters of $\mathcal C_{(q,n,\delta,1)}$ and its dual codes. On the other hand, we give the sufficient and necessary conditions for $\mathcal C_{(q,n,\delta,2)}$ being dually-BCH codes.