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An F q -linear set of rank k , k ≤ h , on a projective line PG (1 , q h) , containing at least one point of weight one, has size at least q k − 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 q k − 1 + 1 in PG (1 , q h) 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 q k − 1 + 1 arise from our construction. Finally, we modify our construction to define rank k linear sets of size q k − 1 + q k − 2 + ... + q k − l + 1 in PG (l , q h). This leads to new infinite families of small minimal blocking sets which are not of Rédei type. [ABSTRACT FROM AUTHOR]