On the linearity of higher-dimensional blocking sets

A small minimal k-blocking set B in PG(n,q), q = p(t), p prime, is a set of less than 3(q(k)+1)/2 points in PC(n,q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG(n,q) are linear over a subfield F(pe) of F(q). Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for k-blocking sets in PG(n,p(t)), with exponent e and p(e) >= 7, it is sufficient to prove it for one value of n that is at least 2k. Further more, we show that the linearity of small minimal blocking sets in PG(2,q) implies the linearity of small minimal k-blocking sets in PG(n,p(t)), with exponent epsilon, with p(e) >= t/epsilon + 11

On sets without tangents and exterior sets of a conic

A set without tangents in \PG(2,q) is a set of points S such that no line meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set of points \E such that all secant lines of \E are external lines of $\mathcal{C}$. In this paper, we first recall some known examples of sets without tangents and describe them in terms of determined directions of an affine pointset. We show that the smallest sets without tangents in \PG(2,5) are (up to projective equivalence) of two different types. We generalise the non-trivial type by giving an explicit construction of a set without tangents in \PG(2,q), $q=p^h$, $p>2$ prime, of size $q(q-1)/2-r(q+1)/2$, for all $0\leq r\leq (q-5)/2$. After that, a different description of the same set in \PG(2,5), using exterior sets of a conic, is given and we investigate in which ways a set of exterior points on an external line $L$ of a conic in \PG(2,q) can be extended with an extra point $Q$ to a larger exterior set of $\mathcal{C}$. It turns out that if $q=3$ mod 4, $Q$ has to lie on $L$, whereas if $q=1$ mod 4, there is a unique point $Q$ not on $L$

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

A note on blockers in posets

The blocker $A^{*}$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done: 1. The posets $P$ for which $A^{**}=A$ for all antichains are characterized. 2. The blocker $A^*$ of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in PG(3, q)

Let Q(+)(3, q) be a hyperbolic quadric in PG(3, q) and T-1 be the set of all lines of PG(3, q) meeting Q(+)(3, q) in singletons (the so-called outer tangents). If k is the minimum size of a T-1-blocking set in PG(3, q), then we prove that k >= q(2) - 1. It is known that there is no T-1-blocking set of size q(2) - 1 for q > 2 even and that there is a unique (up to isomorphism) T-1-blocking set of size 3 for q = 2. For q = 3, we prove as well that there is a unique T-1-blocking set of size 8. Using a computer, we also classify all T-1-blocking sets of size q(2) - 1 for each prime power q <= 13. On basis of some structural similarities we are subsequently able to recognize three families of blocking sets whose further study shows that they can be constructed from certain objects related to finite fields (like nice subsets or permutations of the latter). This connection with finite fields allows us to obtain some computer free descriptions. (C) 2018 Elsevier Inc. All rights reserved

A characterization of Hermitian varieties as codewords

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $PG(r,q^2)$ of the same size as a non-singular Hermitian variety of $PG(r,q^2)$, having the same intersection sizes with the hyperplanes of $PG(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $PG(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $PG(3,q^2)$, $q=p^{h}$, as well as in $PG(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$

A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case