On the Ding and Helleseth's 8th open problem about optimal ternary cyclic codes

The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems due to the efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd, 6th and 7th problems were completely solved, the 3rd, 8th and 9th problems were incompletely solved. In this manuscript, we focus on the 8th problem. By determining the root set of some special polynomials over finite fields, we present a counterexample and a sufficient condition for the ternary cyclic code $\mathcal{C}_{(1, e)}$ optimal. Furthermore, basing on the properties of finite fields, we construct a class of optimal ternary cyclic codes with respect to the Sphere Packing Bound, and show that these codes are not equivalent to any known codes.