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 $\mathcal{C} := {(\text{Tr}(d_1x), \text{Tr}(d_2x), \dots, \text{Tr}(d_nx)) : x \in \mathbb{F}}$ 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.