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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.19141 |
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| _version_ | 1866909551229403136 |
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| author | Cheng, Kaimin Sheng, Du |
| author_facet | Cheng, Kaimin Sheng, Du |
| contents | Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = ϕ(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield. Denote by $\mathrm{Tr}$ the trace function from $\mathbb{F}_q$ to $\mathbb{F}_p$. For $α\in \mathbb{F}_p$ and $β\in \mathbb{F}_q$, let $D$ be the set of nonzero solutions in $\mathbb{F}_q$ to the equation $\mathrm{Tr}(x^{\frac{q-1}{N}} + βx) = α$. Writing $D = \{d_1, \ldots, d_n\}$, we define the code $\mathcal{C}_{α,β} = \{(\mathrm{Tr}(d_1 x), \ldots, \mathrm{Tr}(d_n x)) : x \in \mathbb{F}_q\}$. In this paper, we investigate the weight distribution of $\mathcal{C}_{α,β}$ for all $α\in \mathbb{F}_p$ and $β\in \mathbb{F}_q$, with a focus on general odd primes $p$. When $β= 0$, we establish that $\mathcal{C}_{α,0}$ is a two-weight code for any $α\in \mathbb{F}_p$ and compute its weight distribution. For $β\neq 0$, we determine all possible weights of codewords in $\mathcal{C}_{α,β}$, demonstrating that it has at most $p+1$ distinct nonzero weights. Additionally, we prove that the dual code $\mathcal{C}_{0,0}^{\perp}$ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weight distribution of a class of $p$-ary codes Cheng, Kaimin Sheng, Du Cryptography and Security 94B05 Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = ϕ(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield. Denote by $\mathrm{Tr}$ the trace function from $\mathbb{F}_q$ to $\mathbb{F}_p$. For $α\in \mathbb{F}_p$ and $β\in \mathbb{F}_q$, let $D$ be the set of nonzero solutions in $\mathbb{F}_q$ to the equation $\mathrm{Tr}(x^{\frac{q-1}{N}} + βx) = α$. Writing $D = \{d_1, \ldots, d_n\}$, we define the code $\mathcal{C}_{α,β} = \{(\mathrm{Tr}(d_1 x), \ldots, \mathrm{Tr}(d_n x)) : x \in \mathbb{F}_q\}$. In this paper, we investigate the weight distribution of $\mathcal{C}_{α,β}$ for all $α\in \mathbb{F}_p$ and $β\in \mathbb{F}_q$, with a focus on general odd primes $p$. When $β= 0$, we establish that $\mathcal{C}_{α,0}$ is a two-weight code for any $α\in \mathbb{F}_p$ and compute its weight distribution. For $β\neq 0$, we determine all possible weights of codewords in $\mathcal{C}_{α,β}$, demonstrating that it has at most $p+1$ distinct nonzero weights. Additionally, we prove that the dual code $\mathcal{C}_{0,0}^{\perp}$ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime $p$. |
| title | Weight distribution of a class of $p$-ary codes |
| topic | Cryptography and Security 94B05 |
| url | https://arxiv.org/abs/2503.19141 |