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.     

Recent results on designs with classical parameters

We report on recent results concerning designs with the same parameters as the classical geometric designs PG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ???d ???n?1. The corresponding case of designs with the same parameters as the classical geometric designs AG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.     

Correction to: “Exponential bounds on the number of designs with affine parameters”

It is well known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n?3, grows exponentially. This result was extended recently [5] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2?d?n ? 1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n ? 1. Exponential bounds are also proved for the number of resolvable designs with these parameters. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:156‐166, 2011     

A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks

Consider an incidence structure whose points are the points of a PG_n(n+2,q) and whose block are the subspaces of codimension two, where n?2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n=2 and obtains a Dembowski-Wagner-type result for the class of all such quasi-symmetric designs.     

Exponential bounds on the number of designs with affine parameters

It is well‐known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n≥3, grows exponentially. This result was extended recently [D. Jungnickel, V. D. Tonchev, Des Codes Cryptogr, published online: 23 May, 2009] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2≤d≤n−1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n−1, as well as resolvable designs with these parameters. An exponential lower bound is also proved for the number of non‐isomorphic resolvable 3‐designs with the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, 2), for any 2≤d≤n−1. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 475–487, 2010     

Affine geometry designs, polarities, and Hamada's conjecture

In a recent paper, two of the authors used polarities in PG(2d−1,p) (p?2 prime, d?2) to construct non-geometric designs having the same parameters and the same p-rank as the geometric design PGd(2d,p) having as blocks the d-subspaces in the projective space PG(2d,p), hence providing the first known infinite family of examples where projective geometry designs are not characterized by their p-rank, as it is the case in all known proven cases of Hamada's conjecture. In this paper, the construction based on polarities is extended to produce designs having the same parameters, intersection numbers, and 2-rank as the geometric design AGd+1(2d+1,2) of the (d+1)-subspaces in the binary affine geometry AG(2d+1,2). These designs generalize one of the four non-geometric self-orthogonal 3-(32,8,7) designs of 2-rank 16 (V.D. Tonchev, 1986 [12]), and provide the only known infinite family of examples where affine geometry designs are not characterized by their rank.     

A Hamada type characterization of the classical geometric designs

The dimension of a combinatorial design ${{\mathcal D}}$ over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of ${{\mathcal D}}$ as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design ${{\mathcal D}}$ that has the same parameters as the complement of a classical point-hyperplane design PG _n-1(n, q) or AG _n-1(n, q) is greater than or equal to n + 1, with equality if and only if ${{\mathcal D}}$ is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PG _d (n, q), where 1 ≤ d ≤ n ? 1, in terms of associated codes defined over some extension field E?=?GF(q ^t ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AG _d (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG _d (n, q) for d = 1 and for d > (n ? 2)/2.     

Polarities,quasi-symmetric designs,and Hamada’s conjecture

We prove that every polarity of PG(2k − 1,q), where k≥ 2, gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, PG _k (2k,q). In particular, the case k = 2 yields a new family of quasi-symmetric designs. We also show that our construction provides an infinite family of counterexamples to Hamada’s conjecture, for any field of prime order p. Previously, only a handful of counterexamples were known.      

Lower bounds on covering codes via partition matrices

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ₂(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}.     

Designs having the parameters of projective and affine spaces

Two constructions are described that yield an improved lower bound for the number of 2-designs with the parameters of PG _d (n, q), and a lower bound for the number of resolved 2-designs with the parameters of AG _d (n, q).     

On the classification of semifield flocks

A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF(qⁿ), q odd, with the property that f(x) is a non-zero square for all x∈GF(q). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qⁿ) for q?4n²−8n+2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG(3,qⁿ) for those q exceeding this bound are the linear flocks and the Kantor-Knuth semifield flocks.     

A generalization of a result of Dembowski and Wagner

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG _n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG _d (n, q) with 2 ≤ d ≤  n − 1. We establish this conjecture, hereby improving various previous results.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Automorphisms and Isomorphisms of Symmetric and Affine Designs

Given a finite group G, for all sufficiently large d and for each q > 3 there are symmetric designs and affine designs having the same parameters as PG(d, q) and AG(d, q), respectively, and having full automorphism group isomorphic to G.     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

Some characterizations of quasi-symmetric designs with a spread

The designPG ₂ (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenDPG ₂ (4,q).     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

A trivalent graph with 58 vertices and girth 9

The concept of a partial three-space is due to Laskar and Dunbar, and is a three-dimensional analogue of a partial geometry. Here we determine all partial three-spaces S for which the S-planes are planes of PG(n, q), for which the S-lines are all the lines contained in the S-planes, for which the S-points are all the points in the S-planes, and for which the incidence relation is that of PG(n, q). More generally, we determine all partial three-spaces S for which the S-lines are lines of PG(n,q), for which the S-points are all the points on these lines, and for which the incidence relation is that of PG(n, q).     

Adjacency matrices of polarity graphs and of other C 4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G _q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C ₄) the maximum number of edges of a graph on n vertices and free of squares C ₄. We use the constructed graphs G _q to obtain lower bounds on the extremal function ex(n; C ₄), for some n < q ² + q + 1. In particular, we construct a C ₄-free graph on ${n=q^2 -\sqrt{q}}$ vertices and ${\frac{1}{2} q(q^2-1)-\frac{1}{2}\sqrt{q} (q-1) }$ edges, for a square prime power q.