Some binary BCH codes with length $n=2^m+1$

TL;DR

This paper investigates the dimensions of specific BCH codes with length $n=2^m+1$, employing new techniques to determine coset leaders and code dimensions for certain parameter ranges, advancing understanding of these codes.

Contribution

The paper introduces new techniques for analyzing coset leaders and determines the dimensions of BCH codes with length $n=2^m+1$ for specific values of $m$, which was previously less understood.

Findings

Determined the first five largest coset leaders for certain $m$ values.

Calculated the dimensions of BCH codes with designed distance $ ext{delta}>2^{ ext{ceil}(m/2)}$.

Provided methods that may be applicable to other cyclic codes over finite fields.

Abstract

Under research for near sixty years, Bose-$\!$Ray-$\!$Chaudhuri-$\!$Hocquenghem(BCH) codes have played increasingly important roles in many applications such as communication systems, data storage and information security. However, the dimension and minimum distance of BCH codes are seldom solved until now because of their intractable characteristics. The objective of this paper is to study the dimensions of some BCH codes of length $n=2^m+1$ with $m=2t+1$, $4t+2$, $8t+4$ and $m\geq 10$. Some new techniques are employed to investigate coset leaders modulo $n$. For each type of $m$ above, the first five largest coset leaders modulo $n$ are determined, the dimension of some BCH codes of length $n$ with designed distance $\delta>2^{\lceil \frac{m}{2} \rceil}$ is presented. These new techniques and results may be helpful to study other families of cyclic codes over finite fields.