Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes $\mathcal{C}_{(1,e)}$ where $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ and $a$ is odd. For $a\equiv 3\pmod{4}$, we present two new families of optimal codes with parameters $[3^m-1,3^m-1-2m,4]$, generalizing known constructions. For $a\equiv 1\pmod{4}$, we obtain several nonexistence results on optimal codes $\mathcal{C}_{(1,e)}$ with the aforementioned parameters revealing the constraints of such codes.

翻译：循环码是线性码的一个重要子类，在通信系统和数据存储系统中具有广泛应用。2013年，Ding和Helleseth提出了关于最优三元循环码$\mathcal{C}_{(1,e)}$的九个公开问题。虽然前两个问题和第六个问题已得到完全解决，但其他问题仍未解决。本文通过给出第三和第四个问题的首个反例，并在特定条件下构造了两类最优码，从而部分解决了第三个问题，推进了对这两个问题的研究。此外，我们研究了满足$e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$且$a$为奇数的循环码$\mathcal{C}_{(1,e)}$。对于$a\equiv 3\pmod{4}$，我们提出了两类新的参数为$[3^m-1,3^m-1-2m,4]$的最优码，推广了已知构造。对于$a\equiv 1\pmod{4}$，我们得到了具有上述参数的最优码$\mathcal{C}_{(1,e)}$的若干不存在性结果，揭示了此类码的约束条件。