Intersection distribution of degree four polynomials over finite fields

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2024-10-04 DOI:10.1007/s10623-024-01488-6

Shuxing Li, Maosheng Xiong

引用次数: 0

Abstract

Given a polynomial f over the finite field \(\mathbb {F}_q\), its intersection distribution provides fruitful information on how lines in the affine plane intersect the graph of f over \(\mathbb {F}_q\). The intersection distribution in its simplest cases gives rise to oval polynomials in finite geometry and Steiner triple systems in design theory. Previously, the intersection distribution of degree two and degree three polynomials has been computed. In this paper, we determine the intersection distribution of all degree four polynomials over finite fields. As an application, we present a direct construction of Steiner systems using polynomials with prescribed intersection distribution.

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有限域上四度多项式的交集分布

给定有限域 \(\mathbb {F}_q\)上的多项式 f，它的交点分布提供了关于仿射平面上的线如何与 \(\mathbb {F}_q\)上的 f 的图相交的有用信息。在最简单的情况下，交点分布产生了有限几何中的椭圆多项式和设计理论中的斯坦纳三重系统。在此之前，人们已经计算了二度和三度多项式的交集分布。在本文中，我们确定了有限域上所有四度多项式的交集分布。作为应用，我们提出了一种使用具有规定交集分布的多项式直接构建斯坦纳系统的方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.