On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q ^h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q ^h ).     

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 Rößing 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 condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Multiple blocking sets and multisets in Desarguesian planes

In AG(2, q ²), the minimum size of a minimal (q ? 1)-fold blocking set is known to be q ³ ? 1. Here, we construct minimal (q ? 1)-fold blocking sets of size q ³ in AG(2, q ²). As a byproduct, we also obtain new two-character multisets in PG(2, q ²). The essential idea in this paper is to investigate q ³-sets satisfying the opposite of Ebert’s discriminant condition.     

On sets of type (q + 1, n)2 in finite three-dimensional projective spaces

It is proved that a k–set of type (q + 1, n)₂ in PG(3, q) either is a plane or it has size k ≥ (q + 1)² and a characterization of some sets of size (q + 1)² is given.     

Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties

In Mazzocca et al. (Des. Codes Cryptogr. 44:97–113, 2007), large minimal blocking sets in PG(3, q ²) and PG(4, q ²) have been constructed starting from ovoids of PG(3, q), Q(4, q) and Q(6, q). Some of these can be embedded in a Hermitian variety as maximal partial ovoids. In this paper, the geometric conditions assuring these embeddings are established.     

Projective k-arcs and 2-level secret-sharing schemes

Motivated by applications to 2-level secret sharing schemes, we investigate k-arcs contained in a (q + 1)-arc Γ of PG(3, q), q even, which have only a small number of focuses on a real axis of Γ. Doing so, we also investigate hyperfocused and sharply focused arcs contained in a translation oval of PG(2, q).     

Finite projective geometries that are incidence structures of caps

A t-cap in a geometry is a set of t points no three of which are collinear. A quadric in a projective geometry PG(n, q) is a set of points x which satisfy x^TQx = 0,Q being an n + 1 by n + 1 matrix over GF(q). We show that in any Desarguesian projective geometry PG(2n, q), q odd, there exists a family of (q + 1)-caps which obey the axioms for the lines in a projective geometry, and thus PG(2n, q) can be regarded as an incidence structure of caps. To arrive at this result, a special family of quadrics is considered, the intersections of whose members provide the caps in question.     

Some big sets of mutually disjoint subgeometries of PG(d, q t )

In PG(d, q ^t ) we construct a set ? of mutually disjoint subgeometries isomorphic to PG(d, q) almost partitioning the point set of PG(d, q ^t ) such that there is a group of collineations of PG(d, q ^t ) operating simultaneously as a Singer cycle on all elements of ?. In PG(t?1,q ^t ) we construct big subsets ? of ? whose elements are far away from each other in the following sense:

u

? If P ₁, P ₂ ∈ ? _k , then no point of P ₁ lies on ak-dimensional subspace of P ₂.

For example, we get a set ofq - 1 subplanes of orderq of PG(2,q ³) such that no point of one subplane lies on a line of another subplane, and such that no three points of three different subplanes are collinear.     

A characterisation of tangent subplanes of PG(2, q 3)

In “Barwick and Jackson (Finite Fields Appl. 18:93–107 2012)”, the authors determine the representation of Order-q-subplanes s and order-q-sublines of PG(2, q ³) in the Bruck–Bose representation in PG(6, q). In particular, they showed that an Order-q-subplanes of PG(2, q ³) corresponds to a certain ruled surface in PG(6, q). In this article we show that the converse holds, namely that any ruled surface satisfying the required properties corresponds to a tangent Order-q-subplanes of PG(2, q ³).     

On small blocking sets and their linearity

We prove that a small minimal blocking set of PG(2,q) is “very close” to be a linear blocking set over some subfield GF(pe)<GF(q). This implies that (i) a similar result holds in PG(n,q) for small minimal blocking sets with respect to k-dimensional subspaces (0?k?n) and (ii) most of the intervals in the interval-theorems of Sz?nyi and Sz?nyi-Weiner are empty.     

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

Finite prime-field characteristic sets for planar configurations

Examples are constructed of planar matroids with finite prime-field characteristic sets (i.e. matroids representable over a finite set of prime fields but over fields of no other characteristic). In particular, for any n>3, a projectively unique integer matrix is constructed with 2lsqblog₂nrsqb+6 columns which often gives nonsingleton characteristic sets and, when n is prime, has characteristic set {n}. Many finite subsets of primes are shown to be characteristic sets, including {23,59} (the smallest pair found using these methods), all pairs of primes {p, p′:67?p<p′?293}, and the seventeen largest five-digit primes. Probabilistic arguments are presented to support the conjecture that prime-field characteristic sets exist of every finite cardinality. For p>3, AG(2,p) is shown to be a subset of PG(2, q) only for q=p^s. Another general construction technique suggests that when $\text{P}\text{={p}{}_1\text{,…,p}{}_{\mathrm{k}}\text{}}$ are the primitive prime divisors of 2ⁿ±1 (n sufficiently large), then there is a matroid with O (log n) points whose characteristic set is $\text{P}$. We remark that although only one finite nonsingleton characteristic set (due to R. Reid) was known prior to this paper, a new technique by J. Kahn has shown that every finite set of primes forms a (non-prime-field) characteristic set.     

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.     

Conics and caps

In this article, we begin with arcs in PG(2, q ⁿ ) and show that they correspond to caps in PG(2n, q) via the André/Bruck?CBose representation of PG(2, q ⁿ ) in PG(2n, q). In particular, we show that a conic of PG(2, q ⁿ ) that meets ?_?? in x points corresponds to a (q ⁿ ?+?1 ? x)-cap in PG(2n, q). If x?=?0, this cap is the intersection of n quadrics. If x?=?1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.     

Mixed partitions and related designs

We define a mixed partition of Π =  PG(d, q ^r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q ²) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q ²) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q ²ⁿ . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q ^r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r ² n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q ^rn ).     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.