On linear sets of minimum size

Abstract

An Fq-linear set of rank k, k≤h, on a projective line PG(1,qh), containing at least one point of weight one, has size at least qk−1+1 (see De Beule and Van De Voorde (2019)). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between k∕2 and k−1. Our construction extends the known examples of linear sets of size qk−1+1 in PG(1,qh) constructed for k=h=4 (Bonoli and Polverino, 2005) and k=h in Lunardon and Polverino (2000). We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small k, we investigate whether all linear sets of size qk−1+1 arise from our construction.Finally, we modify our construction to define rank k linear sets of size qk−1+qk−2+…+qk−l+1 in PG(l,qh). This leads to new infinite families of small minimal blocking sets which are not of Rédei type.