[Paper Review] Several classes of optimal ternary cyclic codes

Lanqiang Li, Li Liu+1 more|arXiv (Cornell University)|Jan 5, 2017

Coding theory and cryptography15 references3 citations

TL;DR

This paper constructs eight new classes of optimal ternary cyclic codes with parameters $[3^m-1, 3^m-1-2m, 4]$ by analyzing the non-existence of solutions to a specific equation over $\mathbb{F}_{3^m}$. It resolves three of nine open problems posed by Ding and Helleseth and advances a fourth, significantly extending prior results in optimal ternary cyclic code construction.

ABSTRACT

Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this paper is to give eight new classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$, according to a result on the non-existence of solutions to a certain equation over $F_{3^m}$. It is worth noticing that some recent conclusions on such optimal ternary cyclic codes are some special cases of our work. More importantly, three of the nine open problems proposed by Ding and Helleseth in [8] are solved completely. In addition, another one among the nine open problems is also promoted.

Motivation & Objective

• To construct new classes of optimal ternary cyclic codes with improved parameters for practical applications in communication and storage systems.
• To resolve open problems in the classification of optimal ternary cyclic codes, particularly those posed by Ding and Helleseth.
• To generalize and extend recent results on optimal ternary cyclic codes by identifying broader structural conditions.
• To provide a theoretical foundation for the existence and non-existence of solutions to a key equation over $\mathbb{F}_{3^m}$, enabling code construction.

Proposed method

• Leveraging a known result on the non-existence of solutions to a specific equation over $\mathbb{F}_{3^m}$ to derive conditions for code optimality.
• Applying finite field theory to analyze the structure of cyclic codes over $\mathbb{F}_3$ with length $3^m - 1$.
• Using algebraic techniques to verify that the designed codes achieve the maximum possible minimum distance for their length and dimension.
• Systematically constructing eight new classes of codes by satisfying the derived non-existence condition.
• Validating the optimality of the codes through bounds on minimum distance and code parameters.
• Mapping the construction to known open problems to demonstrate resolution and advancement.

Experimental results

Research questions

• RQ1 Can new classes of optimal ternary cyclic codes be constructed using non-existence results of equations over $\mathbb{F}_{3^m}$?
• RQ2 Which of the nine open problems on optimal ternary cyclic codes proposed by Ding and Helleseth can be fully resolved?
• RQ3 How do the newly constructed codes generalize or extend previous results in the literature?
• RQ4 Can the non-existence of solutions to a specific equation over $\mathbb{F}_{3^m}$ be used as a sufficient condition for code optimality?
• RQ5 What is the impact of these constructions on the classification and understanding of optimal ternary cyclic codes?

Key findings

• Eight new classes of optimal ternary cyclic codes with parameters $[3^m-1, 3^m-1-2m, 4]$ are constructed.
• Three of the nine open problems posed by Ding and Helleseth are completely resolved by the proposed construction.
• One additional open problem is promoted, indicating progress toward its full resolution.
• The results generalize and subsume several recent findings on optimal ternary cyclic codes as special cases.
• The construction relies on a non-existence result of solutions to a key equation over $\mathbb{F}_{3^m}$, which enables systematic code design.
• The codes achieve the maximum possible minimum distance for their length and dimension, confirming their optimality.