Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes

Jingjun Bao, Hanlin Zou

TL;DR

This work advances the theory of optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$ of length $3^m-1$ by addressing two open problems from Ding and Helleseth, presenting explicit counterexamples and two infinite positive constructions that yield $[3^m-1,3^m-1-2m,4]$ codes under specific congruence and gcd conditions. It then broadens the analysis to exponents $e$ satisfying $e(3^h\pm1)\equiv (3^m-a)/2$ with odd $a$, delivering two new families for $a\equiv 3\pmod{4}$ (one with $e(3^h-1)$ and one with $e(3^h+1)$) and establishing necessary conditions for existence. The paper also derives substantial nonexistence results for $a\equiv 5\pmod{8}$ and $a\equiv 1\pmod{8}$, highlighting the constraints posed by cyclotomic coset structures and irreducibility arguments (often verified via computational tools like Magma). Overall, it provides a comprehensive map of when optimal $[3^m-1,3^m-1-2m,4]$ ternary cyclic codes exist, along with concrete constructions and clear obstructions that inform future investigations and applications in coding theory.

Abstract

Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes $\mathcal{C}_{(1,e)}$ where $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ and $a$ is odd. For $a\equiv 3\pmod{4}$, we present two new families of optimal codes with parameters $[3^m-1,3^m-1-2m,4]$, generalizing known constructions. For $a\equiv 1\pmod{4}$, we obtain several nonexistence results on optimal codes $\mathcal{C}_{(1,e)}$ with the aforementioned parameters revealing the constraints of such codes.

Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes

TL;DR

This work advances the theory of optimal ternary cyclic codes of length by addressing two open problems from Ding and Helleseth, presenting explicit counterexamples and two infinite positive constructions that yield codes under specific congruence and gcd conditions. It then broadens the analysis to exponents satisfying with odd , delivering two new families for (one with and one with ) and establishing necessary conditions for existence. The paper also derives substantial nonexistence results for and , highlighting the constraints posed by cyclotomic coset structures and irreducibility arguments (often verified via computational tools like Magma). Overall, it provides a comprehensive map of when optimal ternary cyclic codes exist, along with concrete constructions and clear obstructions that inform future investigations and applications in coding theory.

Abstract

Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes . While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes where and is odd. For , we present two new families of optimal codes with parameters , generalizing known constructions. For , we obtain several nonexistence results on optimal codes with the aforementioned parameters revealing the constraints of such codes.
Paper Structure (15 sections, 24 theorems, 101 equations, 3 tables)

This paper contains 15 sections, 24 theorems, 101 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ding and Helleseth's open problems and the first two new families of optimal ternary cyclic codes
  4. Counterexamples to Problem \ref{['prob7.9']}
  5. Counterexamples to Problem \ref{['prob7.10']}
  6. Two families of optimal ternary cyclic codes with minimum distance four that satisfy all the conditions of Problem \ref{['prob7.9']}.
  7. The third and fourth new families of optimal ternary cyclic codes
  8. The third family of optimal ternary cyclic codes with minimum distance four
  9. The fourth family of optimal ternary cyclic codes with minimum distance four
  10. Further results on cyclic codes $\mathcal{C}_{(1,e)}$ with $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$
  11. Cyclic codes $\mathcal{C}_{(1,e)}$ with $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ where $a\equiv 5\pmod{8}$
  12. Cyclic codes $\mathcal{C}_{(1,e)}$ with $e(3^h+1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ where $a\equiv 1\pmod{8}$
  13. Concluding remarks
  14. Summary of results of this paper
  15. Cyclic codes $\mathcal{C}_{(1,e)}$ with $e(3^h-1)\equiv \frac{3^m-a}{2}\pmod{3^m-1}$ where $a\equiv 1\pmod{8}$

Key Result

Lemma 2.1

Let $e\notin C_1$ and $|C_e|=m$. The ternary cyclic code $\mathcal{C}_{(1,e)}$ has parameters $[3^m-1,3^m-1-2m,4]$ if and only if the following conditions are satisfied: C1: $e$ is even; C2: $(x+1)^e+x^e+1=0$ has a unique solution $x=1$ over $\mathbb{F}_{3^m}$; and C3: $(x+1)^e-x^e-1=0$ has a unique

Theorems & Definitions (52)

  • Lemma 2.1: DH13
  • Remark 2.2
  • Lemma 2.3: DH13
  • Lemma 2.4: LN97
  • Lemma 2.5
  • proof
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 42 more