Dimensions of some LCD BCH codes

TL;DR

This paper studies the dimensions of LCD BCH codes with specific lengths and large designed distances, providing new formulas and bounds that improve upon existing minimum distance estimates.

Contribution

It determines the dimensions of certain LCD BCH codes with lengths related to q^m+1 and provides sharper lower bounds on their minimum distances.

Findings

Explicit dimension formulas for LCD BCH codes of specified lengths.

Determination of code dimensions for a range of designed distances.

Improved lower bounds on minimum distances compared to BCH bounds.

Abstract

In this paper, we investigate the first few largest coset leaders modulo $\frac{q^m+1}{\lambda}$ where $\lambda\mid q+1$ and $q$ is an odd prime power, and give the dimensions of some LCD BCH codes of length $\frac{q^m+1}{\lambda}$ with large designed distances.We also determine the dimensions of some LCD BCH codes of length $n=\frac{(q^m+1)}{\lambda}$ with designed distances $2\leq \delta \leq \frac{ q^{\lfloor(m+1)/2\rfloor}}{\lambda}+1$, where $ \lambda\mid q+1$ and $1<\lambda<q+1$. The LCD BCH codes presented in this paper have a sharper lower bound on the minimum distance than the BCH bound.