Construction of Minimal Ternary Linear Codes with Dimension $n+2$

TL;DR

This paper presents a new generic construction method for minimal ternary linear codes with dimension m+2, including conditions for minimality and explicit weight enumerators, expanding the understanding of code properties beyond the Ashikhmin-Barg condition.

Contribution

It introduces a novel construction for minimal ternary linear codes with dimension m+2, including a necessary and sufficient minimality condition and explicit weight enumerators.

Findings

New class of minimal ternary linear codes violating Ashikhmin-Barg condition

Complete weight enumerators of these codes are explicitly determined

Construction based on exponential sums enhances code design options

Abstract

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for ternary linear codes with dimension $m+2$ is presented, where $m$ is an integer, and a necessary and sufficient condition for this ternary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal ternary linear codes violating the Ashikhmin-Barg condition are obtained, and then their complete weight enumerators are determined.