Small Minimal Blocking Sets inPG(2, q3)

AbstractWe extend the results of Polverino (1999, Discrete Math., 208/209, 469–476; 2000, Des. Codes Cryptogr., 20, 319–324) on small minimal blocking sets in PG(2,p3 ), p prime, p≥ 7, to small minimal blocking sets inPG (2, q3), q=ph, p prime, p≥ 7, with exponent e≥h. We characterize these blocking sets completely as being blocking sets of Rédei-type

A proof of the linearity conjecture for k-blocking sets in PG(n, p3), p prime

In this paper, we show that a small minimal k-blocking set in PG(n, q3), q = p^h, h >= 1, p prime, p >=7, intersecting every (n-k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking sets in PG(n, p^3), p prime, p >=7, are Fp-linear, proving the linearity conjecture (see [7]) in the case PG(n, p3), p prime, p >= 7

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

Sets of generators blocking all generators in finite classical polar spaces

We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q) and H(2n,q^2), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.Comment: accepted for J. Comb. Theory

Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]

On Hyperfocused Arcs in PG(2,q)

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In this paper it is shown that point orbits under suitable groups of elations are hyperfocused arcs with the significant property of being contained neither in a hyperoval, nor in a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set of minimum size, is introduced: a construction method is provided, together with the classification for size up to 10

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