Related papers: A class of ternary codes with few weights

A class of ternary codes with few weights

URL: http://arxiv.org/abs/2410.04216v1
Date: Sat, 5 Oct 2024 16:15:50 GMT
Title: A class of ternary codes with few weights
Authors: Kaimin Cheng,
Abstract summary: In this paper, we investigate a ternary code $mathcalC$ of length $n$, defined by $mathcalC$ := (textTr) := (textTr(dx), dots, dots, d_n$. Using recent results on explicit evaluations of exponential sums, we determine the Weil bound, and techniques, we show that the dual code of $mathcalC$ is optimal with respect to the Hamming bound.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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.

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