BCH codes over $\gf(q)$ with length $q^m+1$

TL;DR

This paper investigates BCH codes over finite fields with length $q^m+1$, analyzing their parameters and demonstrating that they often have good, sometimes optimal, linear code properties.

Contribution

It provides new analysis of BCH codes over $ ext{GF}(q)$ with length $q^m+1$, especially for even $m$, and identifies many codes with optimal parameters.

Findings

BCH codes with length $q^m+1$ have good parameters.

Many of these BCH codes are optimal linear codes.

The paper advances understanding of BCH code parameters for specific lengths.

Abstract

BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $\gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.