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     

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     

Some new symmetric designs

For q, an odd prime power, we construct symmetric (2q²+2q+1,q²,½q(q-1)) designs having an automorphism group of order q that fixes 2q+1 points. The construction indicates that for each q the number of such designs that are pairwise non-isomorphic is very large.     

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let K _q (n, w, t, d) be the minimum size of a code over Z _q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K _q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2 ^m  + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.     

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.     

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.     

New Results on Codes with Covering Radius 1 and Minimum Distance 2

The minimal cardinality of a q-ary code of length n and covering radius at most R is denoted by K_q(n, R); if we have the additional requirement that the minimum distance be at least d, it is denoted by K_q(n, R, d). Obviously, K_q(n, R, d) K_q(n, R). In this paper, we study instances for which K_q(n,1,2) > K_q(n, 1) and, in particular, determine K₄(4,1,2)=28 > 24=K₄(4,1).Supported in part by the Academy of Finland under grant 100500.     

A family of relative difference sets associated with Hjelmslev structures over finite projective spaces

In this note, we construct a new family of relative difference sets, with parameters n=q^d, m=q^d+...+q+1, k=q^d-1(q^d-1), ₁ =q^d-1(q^d-q^d-1-1) and ₂ =q^d-2(q-1)(q^d-1-1) where q is a prime power and d 2 an integer. The associated symmetric divisible designs admit natural epimorphisms onto the symmetric designs formed by points and hyperplanes in the corresponding projective spaces PG(d,q). As in the theory of Hjelmslev planes, points with the same image can be recognized from having the larger of the two possible joining numbers, and dually. More formally, these symmetric divisible designs are balanced c-H-structures (in the sense of Drake and Jungnickel [2]) with parameters c=q^d-2 (q-1)² and t=q^d-1 (q-1) over PG_d-1(d,q). These are the first examples of balanced non-uniform c-H-structures of type 2; they can be used in known constructions to obtain new balanced c-H-structures (for suitable c) of arbitrary type. In fact, all these results are special cases of a more general construction involving arbitrary difference sets.The author gratefully acknowledges the hospitality of the University of Waterloo and the financial support of NSERC under grant IS-0367.     

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     

Dual of two dimensional Lorentz sequence spaces

In [M. Kato and L. Maligranda, On James and Jordan–von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457–465], it is an open problem to compute the James constant of the dual space of two dimensional Lorentz sequence space d⁽²⁾(w,q). In this paper, we shall determine the dual norm of d⁽²⁾(w,q) and completely compute the James constant of d⁽²⁾(w,q).     

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.     

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.     

Ambient Spaces of Dimensional Dual Arcs

A d-dimensional dual arc in PG(n, q) is a higher dimensional analogue of a dual arc in a projective plane. For every prime power q other than 2, the existence of a d-dimensional dual arc (d 2) in PG(n, q) of a certain size implies n d(d + 3)/2 (Theorem 1). This is best possible, because of the recent construction of d-dimensional dual arcs in PG(d(d + 3)/2, q) of size ^d–1 _i=0 q ⁱ, using the Veronesean, observed first by Thas and van Maldeghem (Proposition 7). Another construction using caps is given as well (Proposition 10).     

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.     

Unipotent Conjugacy in General Linear Groups

Let U(n,q) be the group of upper uni-triangular matrices in GL(n,q), the n-dimensional general linear group over the field of q elements. The number of U(n,q)-conjugacy classes in GL(n,q) is, as a function of q, for fixed n, a polynomial in q with integral coefficients.     

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 Structures of Hermitian Varieties

Given a hermitian variety H(d,q²) and an integer k (d–1)/2, a blocking set with respect to k-subspaces is a set of points of H(d,q²) that meets all k-subspaces of H(d,q²). If H(d,q²) is naturally embedded in PG(d,q²), then linear examples for such a blocking set are the ones that lie in a subspace of codimension k of PG(d,q²). Up to isomorphism there are k+1 non-isomorphic minimal linear blocking sets, and these have different cardinalities. In this paper it is shown for 1 k< (d–1)/2 that all sufficiently small minimal blocking sets of H(d,q²) with respect to k-subspaces are linear. For 1 k< d/2–3, it is even proved that the k+1 minimal linear blocking sets are smaller than all minimal non-linear ones.AMS Classification: 1991 MSC: 51E20, 51E21     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.