A new approach to construct minimal linear codes over $\mathbb{F}_{3}$

TL;DR

This work addresses constructing minimal linear codes over $\mathbb{F}_3$ with dimension $n+1$ using two function-based frameworks on partial spreads: ternary functions and characteristic functions. By leveraging the Walsh transform and carefully chosen parameters, the authors derive weight distributions and establish minimality criteria for the resulting codes, denoting explicit conditions under which the Ashikhmin-Barg ratio fails. Notably, they identify families of codes that violate the Ashikhmin-Barg condition, with a ratio $\frac{\mathrm{wt}_{\min}}{\mathrm{wt}_{\max}} \le \tfrac{1}{3}$. The results broaden the catalog of ternary minimal linear codes and offer avenues for constructing higher-dimensional variants in future work.

Abstract

In this article, we present two new approaches to construct minimal linear codes of dimension $n+1$ over $\mathbb{F}_{3}$ using characteristic and ternary functions. We also obtain the weight distributions of these constructed minimal linear codes. We further show that a specific class of these codes violates Ashikhmin-Barg condition.

Theorems & Definitions (27)

• Lemma 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 3.1
• proof