A class of ternary codes with few weights

TL;DR

This paper constructs a new class of ternary linear codes with few weights using finite field trace functions and exponential sums, and demonstrates their optimality in certain cases.

Contribution

It introduces a novel family of ternary codes with few weights based on algebraic and combinatorial methods, expanding the understanding of code weight distributions.

Findings

Determined the weight distribution of the constructed codes.

Proved the dual code is optimal when a0=a0=0.

Applied exponential sum evaluations and Weil bounds in analysis.

Abstract

Let $\ell^m$ be a power with $\ell$ a prime greater than $3$ and $m$ a positive integer such that $3$ is a primitive root modulo $2\ell^m$. Let $\mathbb{F}_3$ be the finite field of order $3$, and let $\mathbb{F}$ be the $\ell^{m-1}(\ell-1)$-th extension field of $\mathbb{F}_3$. Denote by $\text{Tr}$ the absolute trace map from $\mathbb{F}$ to $\mathbb{F}_3$. For any $\alpha \in \mathbb{F}_3$ and $\beta \in\mathbb{F}$, let $D$ be the set of nonzero solutions in $\mathbb{F}$ to the equation $\text{Tr}(x^{\frac{q-1}{2\ell^m}} + \beta x) = \alpha$. In this paper, we investigate a ternary code $\mathcal{C}$ of length $n$, defined by $\mathcal{C} := \{(\text{Tr}(d_1x), \text{Tr}(d_2x), \dots, \text{Tr}(d_nx)) : x \in \mathbb{F}\}$ when we rewrite $D = \{d_1, d_2, \dots, d_n\}$. Using recent results on explicit evaluations of exponential sums, the Weil bound, and combinatorial techniques, we determine the Hamming weight distribution of the code $\mathcal{C}$. Furthermore, we show that when $\alpha = \beta =0$, the dual code of $\mathcal{C}$ is optimal with respect to the Hamming bound.