Several classes of BCH codes of length $n=\frac{q^{m}-1}{2}$
Mengchen Lian
TL;DR
This work addresses the problem of characterizing BCH codes of length $n=\frac{q^{m}-1}{2}$ by deriving exact dimensions for negacyclic codes with specific designed distances, as well as dimensions for codes with few nonzeros and $b\neq 0$. It extends the analysis to the cyclic case by determining the weight distribution of the extended code $\overline{\mathcal{C}}_{(n,1,\delta,1)}$ and the parameters of its dual, for a range of $\delta$ between $\delta_2$ and $\delta_1$, and by establishing a detailed criterion for when certain codes are dually-BCH. The paper thus provides new, explicit dimension formulas, weight distributions, and duality conditions that enhance the understanding of BCH codes of this half-length, with potential to yield best-known codes in some parameter regimes. Overall, the results advance both the theory of BCH codes and their practical parameter selection for communication and storage systems.
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 $C_{(n,-1,\left\lceil \frac{δ+1}{2}\right\rceil,0)}$ for $δ=a\frac{q^m-1}{q-1},aq^{m-1}-1$($1\leq a <\frac{q-1}{2}$) and $δ=a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$ $(2\mid m,1\leq a+b \leq q-1$,$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1)$. Furthermore, the dimensions of negacyclic BCH codes $C_{(n,-1,δ,0)}$ with few nonzeros and $C_{(n,-1,δ,b)}$ with $b\neq 0$ are settled. For cyclic BCH code, we give the weight distribution of extended code $\overline{C}_{(n,1,δ,1)}$ and the parameters of dual code $C^{\perp}_{(n,1,δ,1)}$, where $δ_2\leq δ\leq δ_1$.