On the Ding and Helleseth's 9th open problem about optimal ternary cyclic codes

2025-11-03

AI Summary

This work addresses Open Problem 9 posed by Ding and Helleseth on optimal ternary cyclic codes. We propose a novel construction method based on analyzing the root sets of specific polynomials over finite fields. By rigorously examining the algebraic structure and root-space distribution of cyclic codes—and combining this with the Sphere Packing bound for optimality verification—we construct two new families of optimal ternary cyclic codes with previously unattained parameters. The resulting code lengths and dimensions extend beyond all known constructions, marking the first substantive progress toward infinitely many parameter classes within this open problem’s framework. Our results partially resolve Open Problem 9 and, more significantly, establish a systematic connection between polynomial root-set characterization and optimal code design. This linkage provides a new paradigm for constructing optimal cyclic codes, advancing both theoretical understanding and practical design methodologies in coding theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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 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

Problem

• Solving Ding and Helleseth's 9th open problem about optimal ternary cyclic codes
• Determining root sets of special polynomials over finite fields
• Constructing optimal ternary cyclic codes using Sphere Packing Bound

Innovation

• Determining root sets of special polynomials
• Constructing optimal ternary cyclic codes
• Applying Sphere Packing Bound criteria