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.     

Line spreads of PG(5, 2)

All line spreads of PG(5, 2) are constructed and classified up to equivalence by exhaustive generation considering the specific properties of the automorphism group, and the participation of the spread lines in the subspaces of dimension 3. There are 131,044 inequivalent spreads. The orders of the automorphism groups preserving the spreads, and the 2‐ranks of the related by Rahilly's construction affine 2‐(64,16,5) designs are also computed. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 90–102, 2009     

Subregular Spreads of PG (5, 2e)

The theory of subregular spreads of PG(3,q) was developed by R.H. Bruck (1969, in “Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514. Univ. of North Carolina Press, Chapel Hill). An extension of these results was provided to the higher-dimensional case by the author (1998, Finite Fields Appl.4, 362–380); however, examples of such spreads were only constructed in PG(5, q) for q odd. In this paper, we give a construction of subregular spreads for PG(5, q), where q is even.     

Orthogonal line packings of PG2m − 1(2)

Baker (Discrete Math., 15 (1976), 205–211) has shown that there exists a packing of the lines of each odd dimensional projective space over the field of two elements as a corollary to a theorem asserting the existence of a 2-resolution of the Steiner quadruple system of planes in an even dimensional affine space over the field of two elements. Two packings are orthogonal if any two of their spreads have at most one line in common. A variation of the previous construction gives alternate packings so that, for example, the existence of orthogonal packings of PG_{2m ? 1}(2) when three does not divide 2m ? 1 can be demonstrated.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Orthogonal packings inPG(5, 2)

Aspread inPG(n, q) is a set of lines which partitions the point set. A packing inPG(n, q) (n odd) is a partition of the lines into spreads. Two packings ofPG(n, q) are calledorthogonal if and only if any two spreads, one from each packing, have at most one line in common. Recently, R. D. Baker has shown the existence of a pair of orthogonal packings inPG(5, 2). In this paper we enumerate all packings inPG(5, 2) having both an automorphism of order 31 and the Frobenius automorphism. We find all pairs of orthogonal packings of the above type and display a set of six mutually orthogonal packings. Previously the largest set of orthogonal packings known inPG(5, 2) was two.     

A 35-set of type (2, 5) in PG(2, 9)

A 35-set of type (2, 5) is constructed in the Desarguesian plane of order nine, which is the first example of a set of type (m, q + m) and (m + q)(q² − q + 1) points, with m = 2, in an odd square order plane.     

The cubic Segre variety in PG(5, 2)

The Segre variety in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U ^# of an r-flat is defined to be

A common characterization of the projective spaces PG(4, n) and PG(5, n)

A common characterization of the projective spaces PG(4, n) and PG(5, n) in terms of finite irreducible planar spaces is given.     

Subsets of PG(n,2) and Maximal Partial Spreads in PG(4,2)

Put θ _n = # {points in PG(n,2)} and φ _n = #{lines in PG(n,2)}. Let ψ be anypoint-subset of PG(n,2). It is shown thatthe sum of L = #{internal lines of ψ} and L′= #{external lines of ψ} is the same for all ψ having the same cardinality:[6pt] Theorem A If k is defined by k = |ψ| ? θ _{n ? 1}, then $$L + L' = \phi _{n - 1} + k(k - 1)/2.$$ (The generalization of this to subsets of PG(n,3) is also obtained.) Let $\mathcal{S}$ be a partial spreadof lines in PG(4,2) and let N denote the number of reguli contained in $\mathcal{S}$ .Use of Theorem A gives rise to a simple proof of:[6pt] Theorem B If $\mathcal{S}$ is maximal then one of the followingholds: (i) $\left| \mathcal{S} \right| = 5,{\text{ }}N = 10;{\text{ }}$ (ii) $\left| \mathcal{S} \right| = 7,{\text{ }}N = 4;{\text{ }}$ (iii) $\left| \mathcal{S} \right| = 9,{\text{ }}N = 4.$ If (i) holds then $\mathcal{S}$ is spread in a hyperplane.It is shown that possibility (ii) is realized by precisely threeprojectively distinct types of partial spread. Explicit examplesare also given of four projectively distinct types of partialspreads which realize possibility (iii). For one of these types,type X, the four reguli have a common line. It isshown that those partial spreads in PG(4,2) of size 9 which arise, by a simple construction, from a spreadin PG(5,2), are all of type X.     

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

New Regular Parallelisms of PG(3, 5)

We construct all regular parallelisms of with automorphisms of order 3. Their number is 8. The two cyclic parallelisms found by Prince are among them. The other six ones are the first examples of noncyclic regular parallelisms and the first examples of regular parallelisms that do not belong to the infinite families of Penttila and Williams.     

A characterization of reguli in PG(3,K) and PG(5,K)

In this note, a regulus of lines in PG(3,K) and a regulus of planes in PG(5,K) are characterized by incidence properties.     

Resolutions of PG(5, 2) with point‐cyclic automorphism group

A t‐(υ, k, λ) design is a set of υ points together with a collection of its k‐subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d‐dimensional projective geometry over GF(q), PG(d, q), is a 2‐(q^d + q^d−1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2‐(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = {R₁, R₂, …, R_s}, where s = (υ − 1)/(k−1) and each R_i consists of υ/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛ^σ = ℛ, then the design is said to be point‐cyclically resolvable. The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point‐cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point‐transitive automorphism group. Furthermore, some necessary conditions for the point‐cyclic resolvability of 2‐(υ, k, 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000     

Minimal Blocking Sets in PG(2, 8) and Maximal Partial Spreads in PG(3, 8)

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q ²–q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.