On the Minimum Distances of Some Families of BCH Codes

TL;DR

This paper determines the exact minimum distances of several families of BCH codes, confirming conjectures and constructing codes with specific parameters to improve understanding of their error-correcting capabilities.

Contribution

The paper explicitly constructs locator polynomials for minimum weight codewords, establishing the minimum distances for various primitive and non-primitive BCH code families.

Findings

Infinite families of BCH codes over F_3 and F_4 with d=δ for δ in {5,6,7,8}

Confirmation that the minimum distance equals the Bose distance for certain BCH codes when m ≡ 0 mod pt

Construction of BCH codes with d=δ=p+1 for specific parameters involving prime p

Abstract

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes with $d=\delta$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $\delta$, and offset $b$, denoted by $\mathbf{C}_{(q, n, \delta, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=\delta$, where $\delta \in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=\delta$, where $2 \le \delta \le q-1$. For $\delta=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=\delta$ for all $m$ satisfying $m \equiv 0 \pmod{pt}$, where $p$ denotes the characteristic of $\mathbb{F}_q$. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ is always equal to its Bose distance $d_B$. Our result confirms this conjecture for the case $m \equiv 0 \pmod{pt}$. For non-primitive BCH codes, we construct a family of BCH codes $\mathbf{C}_{(q,\frac{q^p-1}{\lambda},p+1,1)}$ with $d=\delta=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $\lambda \mid q-1$.