Weight distribution of a class of $p$-ary codes
Kaimin Cheng, Du Sheng
TL;DR
The paper determines the weight distribution of a class of $p$-ary linear codes $\mathcal{C}_{\alpha,\beta}$ defined by a trace condition over $\mathbb{F}_q$ with $q=p^{\phi(N)}$ and $N$ chosen so that $p$ is a primitive root modulo $N$. Using explicit evaluations of binomial Weil sums $S_{2\ell^{m}}(a,b)$, the authors show that $\mathcal{C}_{\alpha,0}$ is two-weight with computed distributions, while $\mathcal{C}_{\alpha,\beta}$ with $\beta\neq 0$ has at most $p+1$ distinct nonzero weights; they also establish that $\mathcal{C}_{0,0}^{\perp}$ is optimal under the sphere-packing bound. The work extends previous results for $p=2,3$ to general odd primes $p$, provides a unified framework for analyzing these classes of codes via Weil sums, and highlights dual-code optimality and potential secret-sharing applications through weight-distribution properties. Overall, the study advances understanding of weight distributions in $p$-ary defining-set codes and clarifies the role of $\beta$ in shaping code spectra.
Abstract
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 $p$.