🤖 AI Summary
This study addresses the lack of a general characterization for BCH codes and LCD cyclic codes of length $n = \lambda(q^m + 1)$, where $\lambda \mid q - 1$. By fully describing the structure of $q$-cyclotomic cosets modulo $n$, the work extends existing results—previously limited to the case $\lambda = 1$—to arbitrary $\lambda$ dividing $q - 1$. Leveraging finite field theory, cyclotomic coset analysis, and algebraic coding techniques, the paper explicitly identifies the largest coset representatives for odd $m$ and establishes necessary and sufficient conditions for a BCH code to be self-dual. Key contributions include precise dimension formulas for several families of BCH codes, an improved lower bound on the minimum distance of certain BCH codes up to $2(\delta+1)$—partially achieving optimality—and an exact enumeration of all LCD cyclic codes of the specified length.
📝 Abstract
BCH codes constitute an important class of cyclic codes, many of which are optimal and have wide applications in communication systems. However, determining their parameters remains a challenging problem. In this paper, we investigate BCH codes and LCD cyclic codes over finite fields $\mathbb{F}_q$ of length $n=\lambda(q^m+1)$, where $m$ is a positive integer and $\lambda\mid q-1$. We begin by analyzing the cyclotomic cosets modulo $n$, establishing the sufficient and necessary conditions for $\gamma$ is a coset leader for any $0\le \gamma<n$ and determining the two largest coset leaders. Based on these, we determine the dimensions of several families of BCH codes, and improve the lower bound on their minimal distances. Notably, some of the codes we constructed are optimal. Additionally, when $m$ is odd, we establish necessary and sufficient conditions for a BCH code of length $n$ to be dually-BCH. Furthermore, we enumerate all LCD cyclic codes of length $n$. Finally, several open problems are proposed for further study.