Several classes of BCH codes of length $n=\frac{q^{m}-1}{2}$

TL;DR

This paper analyzes the dimensions and weight distributions of specific negacyclic and cyclic BCH codes of length (q^m-1)/2, providing new formulas and parameters for these classes of codes.

Contribution

It offers new explicit formulas for the dimensions of negacyclic BCH codes and determines the weight distribution and dual code parameters for certain cyclic BCH codes.

Findings

Dimensions of negacyclic BCH codes are explicitly derived.

Weight distribution of extended cyclic BCH codes is characterized.

Parameters of dual codes are established.

Abstract

BCH codes are an important class of linear codes and find extensive utilization in communication and disk storage systems.This paper mainly analyzes the negacyclic BCH code and cyclic BCH code of length $\frac{q^m-1}{2}$. For negacyclic BCH code, we give the dimensions of $C_{(n,-1,\left\lceil \frac{\delta+1}{2}\right\rceil,0)}$ for $\delta =a\frac{q^m-1}{q-1},aq^{m-1}-1$($1\leq a <\frac{q-1}{2}$) and $\delta =a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$ $(2\mid m,1\leq a+b \leq q-1$,$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1)$. Furthermore, the dimensions of negacyclic BCH codes $C_{(n,-1,\delta,0)}$ with few nonzeros and $C_{(n,-1,\delta,b)}$ with $b\neq 0$ are settled. For cyclic BCH code, we give the weight distribution of extended code $\overline{C}_{(n,1,\delta,1)}$ and the parameters of dual code $C^{\perp}_{(n,1,\delta,1)}$, where $\delta_2\leq \delta \leq \delta_1$.