On binomial Weil sums and an application

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 $$S_N(a,b):=\sum_{x\in\mathbb{F}_q\setminus\{0\}}\chi(ax^{\frac{q-1}{N}}+bx),$$ where $\chi$ 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 ${\rm ord}_{N}(p)=\phi(N)$. 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.