Several new classes of optimal ternary cyclic codes with two or three zeros

TL;DR

This work advances the catalog of optimal ternary cyclic codes by constructing four new classes of codes with two zeros ${\mathcal C}_{(0,1,e)}$ and ${\mathcal C}_{(1,e,s)}$ yielding ${[3^m-1,3^m-\frac{3m}{2}-2,4]}$ and $s=\frac{3^m-1}{2}$, and four additional classes with ${[3^m-1,3^m-2m-1,4]}$ through an analysis of irreducible factors and finite-field equations. Leveraging cyclotomic cosets, minimal polynomials, and the Carlet–Ding–Yuan optimality framework, the authors derive explicit exponent choices (e.g., for even $m$, $e=2\cdot 3^{m-1}-3^{\frac{m}{2}-1}-1$; for $m\equiv0\pmod{4}$, $e=(3^m-1)/2+3^{m/2}+1$) that guarantee the desired dimensions and minimum distance $d=4$, while ruling out codewords of weight $2$ or $3$ via targeted equation analysis. They also present a ${\mathcal C}_{(2,e)}$ class and three ${\mathcal C}_{(1,e)}$ classes for odd $m$, supported by DH criteria and explicit open-problem implications, with concrete numerical examples verifying optimality and inequivalence to known constructions. Overall, the paper significantly broadens the landscape of optimal ternary cyclic codes and provides methods adaptable to related $p$-ary settings.

Abstract

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $α$ be a generator of $\mathbb{F}_{3^m}^*$, where $m$ is a positive integer. Denote by $\mathcal{C}_{(i_1,i_2,\cdots, i_t)}$ the cyclic code with generator polynomial $m_{α^{i_1}}(x)m_{α^{i_2}}(x)\cdots m_{α^{i_t}}(x)$, where ${{m}_{α^{i}}}(x)$ is the minimal polynomial of ${{α}^{i}}$ over ${\mathbb{F}_{3}}$. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(0,1,e)}$ and $\mathcal{C}_{(1,e,s)}$ with parameters $[3^m-1,3^m-\frac{3m}{2}-2,4]$, where $s=\frac{3^m-1}{2}$. In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over $\mathbb{F}_{3^m}$, we present four classes of optimal ternary cyclic codes $\mathcal{C}_{(2,e)}$ and $\mathcal{C}_{(1,e)}$ with parameters $[3^m-1,3^m-2m-1,4]$. We show that our new optimal cyclic codes are inequivalent to the known ones.

Theorems & Definitions (28)

• Lemma 1
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• Lemma 8
• Theorem 1
• Example 1