On two conjectures about the intersection distribution

2020/10/01 by Yubo Li, Li, Yubo, Kangquan Li +3 · 1 citation
Computer Science · Mathematics · #11T06 #94A60 #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.2010.00312

openalex publication_date 2020/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently, S. Li and A. Pott\citeLP proposed a new concept of intersection distribution concerning the interaction between the graph \(x,f(x))~|~x∈\F_q\ of f and the lines in the classical affine plane AG(2,q). Later, G. Kyureghyan, et al.\citeKLP proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over \F_q with q both odd and even. They also proposed several conjectures in \citeKLP. In this paper, we completely solve two conjectures in \citeKLP. Namely, we prove two classes of power functions having intersection distribution: v₀(f)=(q(q-1))/(3),~v₁(f)=(q(q+1))/(2),~v₂(f)=0,~v₃(f)=(q(q-1))/(6). We mainly make use of the multivariate method and QM-equivalence on 2-to-1 mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations.

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• The Intersection Distribution: New Results and Perspectives

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