A class of ternary codes with few weights
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914965689991168 |
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| author | Cheng, Kaimin |
| author_facet | Cheng, Kaimin |
| contents | 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 $α\in \mathbb{F}_3$ and $β\in\mathbb{F}$, let $D$ be the set of nonzero solutions in $\mathbb{F}$ to the equation $\text{Tr}(x^{\frac{q-1}{2\ell^m}} + βx) = α$. 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 $α= β=0$, the dual code of $\mathcal{C}$ is optimal with respect to the Hamming bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_04216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A class of ternary codes with few weights Cheng, Kaimin Cryptography and Security Number Theory 94B05 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 $α\in \mathbb{F}_3$ and $β\in\mathbb{F}$, let $D$ be the set of nonzero solutions in $\mathbb{F}$ to the equation $\text{Tr}(x^{\frac{q-1}{2\ell^m}} + βx) = α$. 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 $α= β=0$, the dual code of $\mathcal{C}$ is optimal with respect to the Hamming bound. |
| title | A class of ternary codes with few weights |
| topic | Cryptography and Security Number Theory 94B05 |
| url | https://arxiv.org/abs/2410.04216 |