On binomial Weil sums and an application

2024/09/20 by Kaimin Cheng, Cheng, Kaimin, Shuhong Gao +1 · 1 citation
Mathematics · #11T24 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.13515

openalex publication_date 2024/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbF_q represent the finite field of order q=p^\rm ord_N(p). For a, b∈\mathbbF_q, we define a binomial exponential sum by S_N(a,b):=∑_x∈\mathbbF_q∖\0\χ(ax^(q-1)/(N)+bx), where χ is the canonical additive character of \mathbbF_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.

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• Several classes of p-ary linear codes with few-weights derived from Weil sums

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