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

2025-06-11

AI Summary

This work resolves Open Problem 8, posed by Ding and Helleseth, concerning optimal ternary cyclic codes. Methodologically, we conduct a deep structural analysis of the root sets of specific polynomials over finite fields, enabling the first explicit construction of a counterexample to the conjectured optimality. We further derive rigorous sufficient conditions under which optimality holds. Building on this, we design and prove the existence of an infinite family of novel ternary cyclic codes achieving the Singleton bound—i.e., attaining maximum possible minimum distance for their length and dimension. These codes are verified to be inequivalent to all previously known optimal ternary cyclic codes. Consequently, our results fully settle a central aspect of the open problem, providing a new constructive paradigm for optimal cyclic codes and advancing both cyclic code theory and optimal code design.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Solves Ding's 8th open problem on optimal ternary cyclic codes
• Determines root sets of special polynomials over finite fields
• Constructs new optimal ternary cyclic codes meeting Sphere Packing Bound

Innovation

• Determining root set of special polynomials
• Constructing optimal ternary cyclic codes
• Using finite fields properties for construction