On linear sets of minimum size

Abstract

An F q -linear set of rank k 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 [5]). 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 [2] and k = h in [12]. 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 linear sets of size q k−1 + q k−2 + . . . + q k−l + 1 in PG(l, q). This leads to new infinite families of small minimal blocking sets which are not of Rédei type.

The weight distributions of linear sets in PG(1,q^5)

“…In this paper, we show that possibility (c) does not occur for k = 5 in PG (1, q 5 ). In other words, we see that all linear sets of size q 4 + 1 in PG(1, q 5 ) arise from the construction of [5].…”

“…We find that the possible sizes for an F 3 -linear set of rank 5 in PG(1, 3 5 ) are 82, 91, 97, 100, 103, 106, 109, 112, 115 and 121.…”

“…F 2 -linear t-clubs have been shown to be equivalent with translation KM-arcs of type 2 t (see [2]). A computer search from [8] (phrased in a coding theoretical setting) already showed that there are no 2-clubs in PG(1, 2 5 ). The main theorem of this paper, together with Theorems 1.4, 1.5, and 1.6 show that this result holds for all q, i.e.…”

“…It is not too hard to determine all possibilities for the weight distribution in PG(1, 2 5 ) and PG (1, 3 5 ) by computer (see also Remark 1.7). Using the GAP package FinInG [1], we found the following.…”

“…Remark 1.8. In [5], the authors construct a large family of F q -linear sets of rank k of size q k−1 + 1 which is the smallest possible size under the hypothesis that there is a point of weight one in the set. A simple counting argument shows that for k = 5, such as set of size q 4 + 1 has either:…”