A spectrum result on minimal blocking sets with respect to the planes of PG(3,q), q odd

This article presents a spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd. We prove that for every integer k in an interval of, roughly, size [q (2)/4, 3q (2)/4], there exists such a minimal blocking set of size k in PG(3, q), q odd. A similar result on the spectrum of minimal blocking sets with respect to the planes of PG(3, q), q even, was presented in Roing and Storme (Eur J Combin 31:349-361, 2010). Since minimal blocking sets with respect to the planes in PG(3, q) are tangency sets, they define maximal partial 1-systems on the Klein quadric Q (+)(5, q), so we get the same spectrum result for maximal partial 1-systems of lines on the Klein quadric Q (+)(5, q), q odd

A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results

The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

A study of (x(q+1),x;2,q)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x <= q(2) - q, not decomposable in the sum of another minihyper and a line, a (j (q + 1), j; 2, q)-minihyper, where j = q(2) - q-x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44: 169-196, 2007), giving further results on these minihypers

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 q(2) of size q(2) + 2q + 2 admitting 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 q(2) of size at most 4q(2)/3 + 5q/3, which is considerably smaller than 2q(2) - 1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q(2). We also consider particular Andre 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