An infinite family of symmetric designs

In this paper, using the construction method of [3], we show that if q>2 is a prime power such that there exists an affine plane of order q?1, then there exists a strongly divisible 2?(q?1)(q^h?1), q^h?1(q?1), q^h?1) design for every h?2. We show that these quasi-residual designs are embeddable, and hence establish the existence of an infinite family of symmetric 2?(q^h+1?q+1,q^h, q^h?1) designs. This construction may be regarded as a generalisation of the construction of [1, Chapter 4, Section 1] and [4].     

A contraction of square transversal designs

In this paper we show that if a square transversal design TD_λ[k;u], say D(=(P,B)), admits a class semiregular automorphism group G of order s, then we have a by matrix M with entries from G∪{0} satisfying , where , if i=j, and , otherwise. As an application of (*), we show that any symmetric TD₂[12;6] admits no nontrivial elation. We also obtain a result that gives us a restriction on the existence of elations of putative projective planes of composite order.     

Halving transversal designs

A factor H of a transversal design TD(k,n) = (V,𝒢, ℬ︁), where V is the set of points, 𝒢 the set of groups of size n and ℬ︁ the set of blocks of size k, is a triple (V,𝒢, 𝒟) such that 𝒟 is a subset of ℬ︁. A halving of a TD (k, n) is a pair of factors H_i = (V, 𝒢, 𝒟_i), i = 1,2 such that 𝒟₁ ∪ 𝒟₂ = ℬ︁, 𝒟₁ ∩ 𝒟₂ = ∅︁ and H₁ is isomorphic to H₂. A path of length q is a sequence x₀, x₁,…,x_q of points such that for each i = 1, 2,…, q the points x_i‐1 and x_i belong to a block B_i and no point appears more than once. The distance between points x and y in a factor H is the length of the shortest path from x to y. The diameter of a connected factor H is the maximum of the set of distances among all pairs of points of H. We prove that a TD (3, n) is halvable into isomorphic factors of diameter d only if d = 2,3,4, or ∞ and we completely determine for which values of n there exists such a halvable TD (3, n). We also show that if any group divisible design with block size at least 3 is decomposed into two factors with the same finite diameter d, then d≤ 4. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 83–99, 2000     

Thwarts in transversal designs

A subset of points in a transversal design is athwart if each block in the design has one of a small number of intersection sizes with the subset. Applications to the construction of mutually orthogonal latin squares are given. One particular case involves inequalities for the minimum number of distinct symbols appearing in an × subarray of an×n latin square. Using thwarts, new transversal designs are determined for orders 408, 560, 600, 792, 856, 1046, 1059, 1368, 2164, 2328, 2424, 3288, 3448, 3960, 3992, 3994, 4025, 4056, 4824, 5496, 6264, 7768, 7800, 8096, and 9336.     

A construction of non self-orthogonal designs

It is known that a self-orthogonal 2-(21,6,4) design is unique up to isomorphism. We give a construction of 2-(21,6,4) designs. As an example, we obtain non self-orthogonal 2-(21,6,4) designs. Furthermore, we also consider a generalization of the construction.     

Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

On simple and supersimple transversal designs

In this paper, the existence of a transversal design TD_λ (4, g) is proved for all indices λ satisfying 2 ≤ λ ≤ g such that any two of its blocks intersect in at most two elements. Similar results are obtained for transversal designs without repeated blocks. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 311–320, 2000     

A non symmetric FVM-BEM coupling method

We consider the prototype model for flow and transport of a concentration in porous media in an interior domain and couple it with a diffusion process in the corresponding unbounded exterior domain. To guarantee mass conservation and stability with respect to dominating convection also for a discrete solution we introduce a non symmetric coupling of the vertex-centered finite volume method (FVM) and the boundary element method (BEM). BEM approximates the unbounded exterior problem which avoids truncation of the domain. One can also interpret that the (unbounded) exterior problem “replaces” the boundary conditions of the interior problem. We aim to provide a first rigorous analysis of the discrete system for different model parameters; existence and uniqueness, convergence, and a priori estimates. Numerical examples illustrate the strength of the chosen method which is computational cheaper than the previous three field FVM-BEM couplings. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-affine symmetric designs

A symmetric design with parameters v = q ²(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P ₀, P ₁,..., P _q , P _q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2. Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India.     

A census of highly symmetric combinatorial designs

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v,k,1) designs, mainly for t=2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t>2 most of these characterizations have remained long-standing challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 3≤t≤6 is of particular interest and has been open for about 40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating back to 1965). The present paper continues the author’s work (see Huber (J. Comb. Theory Ser. A 94, 180–190, 2001; Adv. Geom. 5, 195–221, 2005; J. Algebr. Comb., 2007, to appear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.      

A characterization of the extensions of symmetric designs

Let D be a quasi-symmetric design with block intersection numbers 0 and y. For any fixed block B of D, let D_B denote the incidence structure whose points are the points of D not in B and whose blocks are the blocks of D disjoint from B. If D is an extension of a symmetric design, Cameron showed that D_B is a design and the parameters of D are exactly one of four types. We prove the converse: If D_B is a design, then D is the extension of a symmetric design.     

Imprimitive flag-transitive symmetric designs

A recent paper of O'Reilly Regueiro obtained an explicit upper bound on the number of points of a flag-transitive, point-imprimitive, symmetric design in terms of the number of blocks containing two points. We improve that upper bound and give a complete list of feasible parameter sequences for such designs for which two points lie in at most ten blocks. Classifications are available for some of these parameter sequences.     

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.