On regular sets of affine type in finite Desarguesian planes and related codes

In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines are 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 Szonyi 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 as a unital and meet affine lines of PG(2, q2) in one of 4 possible intersection numbers, each of them congruent to 1 modulus ,/q. As a byproduct, we also determine the intersection sizes of the Hermitian curve defined over GF(q2), q a square, with suitable rational curves of degree ,/q and we obtain ,/q - divisible codes with 5 non -zero weights. We also determine the weight enumerator of the codes arising from the general constructions up to some q -powers.