The Dimension and Minimum Distance of Two Classes of Primitive BCH Codes

TL;DR

This paper determines the dimension and minimum distance of two specific classes of narrow-sense primitive BCH codes, providing new theoretical insights and identifying some as optimal or near-optimal codes.

Contribution

It derives the dimension and minimum distances for two classes of primitive BCH codes with specific design distances, expanding understanding of their parameters.

Findings

Some codes are optimal in terms of error correction.

Other codes are among the best known linear codes.

Weight distributions are also characterized.

Abstract

Reed-Solomon codes, a type of BCH codes, are widely employed in communication systems, storage devices and consumer electronics. This fact demonstrates the importance of BCH codes -- a family of cyclic codes -- in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with design distances $\delta=(q-1)q^{m-1}-1-q^{\lfloor (m-1)/2\rfloor}$ and $\delta=(q-1)q^{m-1}-1-q^{\lfloor (m+1)/2\rfloor}$. The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known.