In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of $\mathrm{PG}(2, q^2)$ in one of $4$ possible intersection numbers, each of them congruent to $1$ modulo $\sqrt{q}$. As a byproduct, we determine the intersection sizes of the Hermitian curve defined over $\mathrm{GF}(q^2)$ with suitable rational curves of degree $\sqrt{q}$ and we obtain $\sqrt{q}$-divisible codes with $5$ non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some $q$-powers.

翻译：本文研究有限Desarguesian平面中的点集，这些点集与直线的交点数多重集在除一条特殊平行直线类外的所有直线上相同。我们将此类集合称为仿射型正则集。当特殊平行直线类中的直线具有相同的交点数时，则称此类集合为点型正则集。经典例子如单位圆；对具有少交点数的此类集合的详细研究与构造可追溯至Hirschfeld和Sz\H{o}nyi在1991年的工作。本文提出了正则集的一些通用构造方法，并描述了若干无限族。其中一个族的集合大小与单位圆相同，且与$\mathrm{PG}(2, q^2)$的仿射直线相交于$4$种可能的交点数之一，每个交点数模$\sqrt{q}$同余于$1$。作为副产品，我们确定了定义在$\mathrm{GF}(q^2)$上的Hermitian曲线与次数为$\sqrt{q}$的适当有理曲线的交集大小，并获得了具有$5$个非零权重的$\sqrt{q}$-可分数码。我们还确定了由一般构造模某些$q$次幂得到的码的权重枚举式。