On binomial Weil sums and an application
Kaimin Cheng, Shuhong Gao

TL;DR
This paper explicitly evaluates binomial Weil sums over finite fields for certain parameters, constructs related linear codes, and proves their optimality, extending previous results to odd characteristic fields.
Contribution
It provides a direct evaluation of binomial Weil sums for odd primes with specific order conditions and constructs optimal ternary linear codes from these sums.
Findings
Explicit formula for $S_N(a,b)$ when ${ m ord}_N(p)=(N)$
Construction of ternary linear codes with known weight distribution
Dual codes are optimal with respect to sphere packing bound
Abstract
Let be a prime, and be a positive integer not divisible by . Denote by the multiplicative order of modulo . Let represent the finite field of order . For , we define a binomial exponential sum by where is the canonical additive character of . In this paper, we provide an explicit evaluation of for any odd prime and any satisfying . 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…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory
