A class of ternary codes with few weights

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 $\mathcal{C}:=\{(\text{Tr}(d_{1}x),\text{Tr}(d_{2}x),\dots ,\text{Tr}(d_{n}x)):x\in \mathbb{F}\}$ 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.

arXiv comment18 pages