WU Tingting, GAO Jian
Cyclic codes are an important subclass of linear codes, which have wide applications in data storage systems, communication systems due to their efficient encoding and decoding algorithms. In this paper, quinary cyclic codes $\mathcal{C}_{(1,e,s)}$ with three nonzeros $\alpha$, $\alpha^e$, $\alpha^s$ are discussed, where $\alpha$ is the primitive element of $\mathbb{F}_{5^m}$, $m$ is a positive integer, $s=\frac{5^m-1}{2}$, and $2\leq e\leq 5^m-2$. Firstly, the paper presents the necessary and sufficient condition for the quinary cyclic codes $\mathcal{C}_{(1,e,s)}$ to be optimal when $e=5^h-2$, where $1\leq h\leq m$. Furthermore, based on the proposed necessary and sufficient condition, by analyzing the irreducible factors of certain polynomials, it is proven that when $m$ is an odd integer no less than $3$, and $h$ takes values of $2,~m-2,~m-1$, respectively, the cyclic codes $\mathcal{C}_{(1,e,s)}$ are optimal with parameters $[5^m-1,5^m-2m-2,4]$. Secondly, when $e=4(5^h+1)$, where $0\leq h\leq m-1$, by analyzing the solutions of certain equations, it is demonstrated that when $m$ is an odd integer no less than $3$ and $h$ is $0$, the cyclic code $\mathcal{C}_{(1,e,s)}$ is optimal with parameters $[5^m-1,5^m-2m-2,4]$. This paper has made some progress in solving the two open problems proposed in reference(Wu, et al., 2023).
