Blocking and Double Blocking Sets in Finite Planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q2 of size q2 + 2q + 2 admitting ∗This author has been supported as a postdoctoral fellow of the Research Foundation Flanders (Belgium) (FWO). †This author was supported by a Visiting Professor grant of the Special Research Fund Ghent University (BOF project number 01T00413) and a Research Grant of the Research Foundation Flanders (Belgium) (FWO) (project number 1504514N). ‡This author is a postdoctoral fellow of the Research Foundation Flanders (Belgium) (FWO). the electronic journal of combinatorics 23(2) (2016), #P2.5 1 1-,2-,3-,4-, (q + 1)and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q2 of size at most 4q2/3+5q/3, which is considerably smaller than 2q2−1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q2. We also consider particular André planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.