BCH codes of length $n=λ(q^m+1)$ over finite fields
This work provides new insights into the parameters and properties of BCH codes, which are important for researchers and practitioners working with error-correcting codes in communication systems.
This paper investigates BCH codes and LCD cyclic codes over finite fields of length $n=λ(q^m+1)$, determining the dimensions of several families of BCH codes and improving the lower bound on their minimal distances. The authors constructed some optimal codes and established conditions for BCH codes to be dually-BCH when $m$ is odd.
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 $n=λ(q^m+1)$, where $m$ is a positive integer and $λ\mid q-1$. We begin by analyzing the cyclotomic cosets modulo $n$, establishing the sufficient and necessary conditions for $γ$ is a coset leader for any $0\le γ<n$ 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.