On binomial Weil sums and an application
Kaimin Cheng, Shuhong Gao
TL;DR
This work provides explicit evaluations of binomial Weil sums $S_N(a,b)$ in odd characteristic under the primitive-root condition ${ord}_N(p)=\phi(N)$ for $N\in\{2,4,\wp^m,2\wp^m\}$, enabling direct computation in terms of $S_N(c)$ and $\sqrt{q}$. The authors develop an elementary approach that yields precise formulas and use these results to construct a two-weight ternary code with fully determined weight distribution, along with an analysis showing the dual codes are optimal with respect to sphere-packing bounds. The key technique combines cyclotomic irreducibility, basis expansions in $\xi$, and Moisio-type evaluations, with a generalization to linearized perturbations. The findings extend prior even-characteristic results to odd characteristic and offer a practical method for generating and analyzing $p$-ary codes from exponential sums, with potential applications in secret sharing, authentication, and combinatorial designs.
Abstract
Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$. Denote by ${\rm ord}_N(p)$ the multiplicative order of $p$ modulo $N$. Let $\mathbb{F}_q$ represent the finite field of order $q=p^{{\rm ord}_N(p)}$. For $a, b\in\mathbb{F}_q$, we define a binomial exponential sum by $$S_N(a,b):=\sum_{x\in\mathbb{F}_q\setminus\{0\}}χ(ax^{\frac{q-1}{N}}+bx),$$ where $χ$ is the canonical additive character of $\mathbb{F}_q$. In this paper, we provide an explicit evaluation of $S_{N}(a,b)$ for any odd prime $p$ and any $N$ satisfying ${\rm ord}_{N}(p)=φ(N)$. Our elementary and direct approach allows for the construction of a class of ternary linear codes, with their exact weight distribution determined. Furthermore, we prove that the dual codes achieve optimality with respect to the sphere packing bound, thereby generalizing previous results from even to odd characteristic fields.