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，这些点集与所有直线（除一条例外平行类直线）的相交数的多重集相同。我们将此类点集称为仿射型正则集。当例外平行类中所有直线具有相同的相交数时，则称这些点集为点尖型正则集。经典实例如unitals；Hirschfeld与Sz\H{o}nyi于1991年对这类具有少量相交数的点集进行了详细研究与构造。本文提供若干正则集的一般构造方法，并描述无穷族系。其中一族中元素具有unital的大小，且与$\mathrm{PG}(2, q^2)$中仿射直线的相交数仅取$4$种可能值，每个值均模$\sqrt{q}$余$1$。作为副产品，我们确定了定义于$\mathrm{GF}(q^2)$上的Hermitian曲线与适当$\sqrt{q}$次有理曲线的相交大小，并得到具有$5$个非零权重的$\sqrt{q}$-可分码。我们还确定了基于模若干$q$幂次的通用构造所得码的权重枚举子。