On the Ding and Helleseth's 9th open problem about optimal ternary cyclic codes
Peipei Zheng, Dong He, Qunying Liao
TL;DR
This paper addresses Ding and Helleseth's 9th open problem on optimal ternary cyclic codes by analyzing the root sets of specific polynomials over finite fields. It presents two counterexamples to the proposed 9th-problem conditions, showing the problem cannot be settled in full in general. It then constructs three classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$ by enforcing and verifying the conditions $Q_1$–$Q_3$ via root-set analysis and congruence constraints on $m$ and $e$, introducing supplementary lemmas (A1–A2) and Theorem (A3) to expand the set of cases where optimality holds. The results advance the understanding of distance-4, sphere-packing-optimal ternary cyclic codes and provide concrete constructions tied to the algebra of cyclotomic cosets and minimal polynomials over $\mathbb{F}_{3^m}$.
Abstract
The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. Till now, the 1st, 2nd and 6th problems were completely solved, and the 3rd, 7th, 8th and 9th problems were partially solved. In this manuscript, we focus on the 9th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer for the 9th problem, and then we construct two classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound basing on some special polynomials over finite fields