Translation hyperovals and $\mathbb{F}_2$-linear sets of pseudoregulus type

关于半域平面中的平移超卵形线

On Translation Hyperovals in Semifield Planes

本文构造了第一个不含平移超卵形线的有限平移平面（广义扭曲域平面，阶64），否定了Cherowitzo猜想，并将其与秩度量码覆盖半径及散射子空间不存在性相关联。

Hyperovals in Knuth's binary semifield planes

In each of the three projective planes coordinatised by the Knuth's binary semifield $\mathbb{K}_n$ of order $2^n$ and two of its Knuth derivatives, we exhibit a new family of infinitely many translation hyperovals. In particular, when $n=5$, we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied.

On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$

Scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$ have been defined and investigated in [G. Lunardon, G. Marino, O. Polverino, R. Trombetti: Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$. J. Algebr. Comb. 39 (2014), 807--831.; G. Donati, N. Durante: Scattered linear sets generated by collineations between pencils of lines. J. Algebr. Comb. 40 (2014), 1121-1134]. The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say $L$, are proved by means of three different ways…

Scattered linear sets in a finite projective line and translation planes

Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f…

有限偶数阶平面上平移椭圆的一个特征

A characterisation of translation ovals in finite even order planes

本文考虑PG（4，q）中的点集C，q为偶数，满足关于PG（4，q）平面的某些组合性质。我们证明了在无穷远处的超平面中存在一个正则扩展，使得在相应的Bruck-Bose平面PG（2，q^2）中，对应于C的点形成一个平移超椭圆，反之亦然。

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a fa…