Minimal 1-saturating sets and complete caps in binary projective spaces

In binary projective spaces PG(v,2), minimal 1-saturating sets, including sets with inner lines and complete caps, are considered. A number of constructions of the minimal 1-saturating sets are described. They give infinite families of sets with inner lines and complete caps in spaces with increasing dimension. Some constructions produce sets with an interesting symmetrical structure connected with inner lines, polygons, and orbits of stabilizer groups. As an example we note an 11-set in PG(4,2) called “Pentagon with center”. The complete classification of minimal 1-saturating sets in small geometries is obtained by computer and is connected with the constructions described.     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3,q), small Rédei minimal blocking sets of PG(2,q²), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface H(3,q²).     

Embedding 1-Factorizations of K n in PG(2, 32)

1-Factorizations of the complete graph K _n embedded in a finite Desarguesian projective plane PG(2, q), q even, are hyperfocused arcs of size n. The classification of hyperfocused arcs is motivated by applications to 2-level secret sharing schemes. So far it has been done for q  ≤ 16, and for special types of hyperfocused arcs. In this paper the case q = 32 is investigated and the following two results are proven. (i) Uniqueness of hyperfocused 12-arcs, up to projectivities. (ii) Non-existence of hyperfocused 14-arcs.     

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.     

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.     

The number of designs with geometric parameters grows exponentially

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PG _n-1(n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PG _d (n, q), where 2 ≤ d ≤ n ? 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters.     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

On 1-Blocking Sets in PG(n,q), n ≥ 3

In this article we study minimal1-blocking sets in finite projective spaces _PG(n,q),n 3. We prove that in PG(n,q ²),q = p ^h , p prime, p > 3,h 1, the second smallest minimal 1-blockingsets are the second smallest minimal blocking sets, w.r.t.lines, in a plane of PG(n,q ²). We also study minimal1-blocking sets in PG(n,q ³), n 3, q = p ^h, p prime, p > 3,q 5, and prove that the minimal 1-blockingsets of cardinality at most q 3 + q ² + q + ¹ are eithera minimal blocking set in a plane or a subgeometry PG(3,q).     

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

In De Beule and Storme, Des Codes Cryptogr 39(3):323–333, De Beule and Storme characterized the smallest blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 4; they proved that these blocking sets are truncated cones over the unique ovoid of Q ⁺(7, 3). We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 3, of size at most 3 ⁿ + 3 ^n–2. This means that the three smallest minimal blocking sets of Q ⁺(2n + 1, 3), n ≥ 3, are now classified. We present similar results for q = 2 by classifying the minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, of size at most 2 ⁿ + 2 ^n-2. This means that the two smallest minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, are classified.     

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.     

Blocking Sets of Certain Line Sets Related to a Conic

In this paper we classify point sets of minimum size of two types (1) point sets meeting all secants to an irreducible conic of the desarguesian projective plane PG(2,q), q odd; (2) point sets meeting all external lines and tangents to a given irreducible conic of the desarguesian projective plane PG(2,q), q even.     

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

On Maximal Partial Spreads in PG(n, q)

In this paper we construct maximal partial spreads in PG(3, q) which are a log q factor larger than the best known lower bound. For n ≥ 5 we also construct maximal partial spreads in PG(n, q) of each size between cnq ^{n ? 2} log q and c′ q ^{n ? 1}.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

Orbits on the projective line

Let PGL(2, q) act in the natural way on the four-sets and the five-sets in PG(1, q). We determine the number of orbits of any given size and use this to construct some 3-designs on q + 2 points.     

Linear (q+1)-fold Blocking Sets in PG(2, q4)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed