2020/03/22 by Gohar Kyureghyan, Kyureghyan, Gohar, Shuxing Li +3 · 1 citation
Computer Science · Mathematics · #Coding theory and cryptography #Analytic Number Theory Research #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2003.10040
The intersection distribution of a polynomial f over finite field\n mathbbFq was recently proposed in Li and Pott (arXiv:2003.06678v1), which\nconcerns the collective behaviour of a collection of polynomials f(x)+cx\n\| c \∈ mathbbFq . The intersection distribution has an underlying\ngeometric interpretation, which indicates the intersection pattern between the\ngraph of f and the lines in the affine plane AG(2,q). When q is even, the\nlong-standing open problem of classifying o-polynomials can be rephrased in a\nsimple way, namely, classifying all polynomials which have the same\nintersection distribution as x2. Inspired by this connection, we proceed to\nconsider the next simplest case and derive the intersection distribution for\nall degree three polynomials over mathbbFq with q both odd and even.\nMoreover, we initiate to classify all monomials having the same intersection\ndistribution as x3, where some characterizations of such monomials are\nobtained and a conjecture is proposed. In addition, two applications of the\nintersection distributions of degree three polynomials are presented. The first\none is the construction of nonisomorphic Steiner triple systems and the second\none produces infinite families of Kakeya sets in affine planes with previously\nunknown sizes.\n