Semifields, relative difference sets, and bent functions
Alexander Pott, Kai-Uwe Schmidt, Yue Zhou
备注：
Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyter
Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.
Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$
本文研究了射影空间$PG(2n-1,q^t)$（$n \geq 1$，$t\geq 3$）的一类离散$\F_q$ --秩为$tn$的线性集，称为{ \it伪正则型}，推广了[G.马里诺岛波尔韦里诺河Trombetti：On ${\mathbb F}_q$ --线性集合的$\PG(3,q^3)$和半域，{ \em J. Combin. Theory Ser. A}，{ \bf 114}（2007），769- 788]和[M.拉夫劳湾Van de Voorde：Scattered linear sets and pseudoreguli，preprint].作为应用，我们用相关的线性集刻画了一些经典的半域族：广义扭域和二维Knuth半域。
Cyclotomic skew partial difference sets and partial difference families
School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）；Institut für Mathematik, Alpen-Adria-Universität Klagenfurt（克拉根福阿尔卑斯-亚得里亚大学数学研究所）
