Constructions for Circulant and Group‐Developed Generalized Weighing Matrices

An elementary construction yields a new class of circulant (so‐called “Butson‐type”) generalized weighing matrices, which have order and weight n², all of whose entries are nth roots of unity, for all positive integers , where . The idea is extended to a wider class of constructions giving various group‐developed generalized weighing matrices.     

Hadamard Matrices Modulo 5

In this paper, we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5‐modular Hadamard matrices of order n if and only if or . In particular, this solves the 5‐modular version of the Hadamard conjecture.     

Hadamard Matrices Modulo p and Small Modular Hadamard Matrices

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7‐modular and 11‐modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p‐modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p‐modular version of the Hadamard conjecture for all but finitely many cases when , and prove a weaker result for . Finally, we look at constraints on the existence of m‐modular Hadamard matrices when the size of the matrix is small compared to m.     

On Skew E–W Matrices

An E–W matrix M is a ( ? 1, 1)‐matrix of order , where t is a positive integer, satisfying that the absolute value of its determinant attains Ehlich–Wojtas' bound. M is said to be of skew type (or simply skew) if is skew‐symmetric where I is the identity matrix. In this paper, we draw a parallel between skew E–W matrices and skew Hadamard matrices concerning a question about the maximal determinant. As a consequence, a problem posted on Cameron's website [7] has been partially solved. Finally, codes constructed from skew E–W matrices are presented. A necessary and sufficient condition for these codes to be self‐dual is given, and examples are provided for lengths up to 52.     

Twofold triple systems with cyclic 2‐intersecting Gray codes

Given a combinatorial design with block set , the block‐intersection graph (BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . The i‐block‐intersection graph (i‐BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . In this paper, several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2‐BIGs and result in larger TTSs that also have Hamiltonian 2‐BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2‐BIGs (equivalently TTSs with cyclic 2‐intersecting Gray codes) as well as the complete spectrum for TTSs with 2‐BIGs that have Hamilton paths (i.e. for TTSs with 2‐intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes.     

On ‐Cocyclic Hadamard Matrices

A characterization of ‐cocyclic Hadamard matrices is described, depending on the notions of distributions, ingredients, and recipes. In particular, these notions lead to the establishment of some bounds on the number and distribution of 2‐coboundaries over to use and the way in which they have to be combined in order to obtain a ‐cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in A. Baliga, K. J. Horadam, Australas. J. Combin., 11 (1995), 123–134 is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ‐cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, define representatives for them, and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way, in terms of diagrams. Let be the set of cocyclic Hadamard matrices over having a symmetric diagram. We also prove that the set of Williamson‐type matrices is a subset of of size .     

New Results on D‐Optimal Matrices

We construct a number of new supplementary difference sets (SDS) with v odd and . In particular, these give rise to D‐optimal matrices of the four new orders 206, 242, 262, 482, constructed here for the first time.     

On the Choice Number of Packings

In this note, we show that for positive integers s and k, there is a function such that every t‐ packing with at least edges, , has choice number greater than s. Consequently, for integers s, k, t, and λ there is a such that every t‐ design with has choice number greater than s. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 504‐507, 2012     

Optimal 1—Spontaneous Emission Error Designs*

A t‐spontaneous emission error design, denoted by t‐ SEED or t‐SEED in short, is a system of k‐subsets of a v‐set V with a partition of satisfying for any and , , where is a constant depending only on E. The design of t‐SEED was introduced by Beth et al. in 2003 (T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, M. Mussinger, Des Codes Cryptogr 29 (2003), 51–70) to construct quantum jump codes. The number m of designs in a t‐ SEED is called dimension, which corresponds to the number of orthogonal basis states in a quantum jump code. A t‐SEED is nondegenerate if every point appears in each of its member design. A nondegenerate t‐SEED is called optimal when it achieves the largest possible dimension. This paper investigates the dimension of optimal 1‐SEEDs, in which Baranyai's Lemma plays a significant role and the hypergraph distribution is closely related as well. Several classes of optimal 1‐SEEDs are shown to exist. In particular, we determine the exact dimensions of optimal 1‐ SEEDs for all orders v and block sizes k with .     

Cycle Extensions in BIBD Block‐Intersection Graphs

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non‐Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD, consists of a set V of v elements and a block set of k‐subsets of V such that each 2‐subset of V occurs in exactly λ of the blocks of . The block‐intersection graph of a design is the graph having as its vertex set and such that two vertices of are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block‐intersection graph of any BIBD is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block‐intersection graph.     

Degree‐ and Orbit‐Balanced Γ‐Designs When Γ Has Five Vertices

A Γ‐design of the complete graph is a set of subgraphs isomorphic to Γ (blocks) whose edge‐sets partition the edge‐set of . is balanced if the number of blocks containing x is the same number of blocks containing y for any two vertices x and y. is orbit‐balanced, or strongly balanced, if the number of blocks containing x as a vertex of a vertex‐orbit A of Γ is the same number of blocks containing y as a vertex of A, for any two vertices x and y and for every vertex‐orbit A of Γ. We say that is degree‐balanced if the number of blocks containing x as a vertex of degree d in Γ is the same number of blocks containing y as a vertex of degree d in Γ, for any two vertices x and y and for every degree d in Γ. An orbit‐balanced Γ‐design is also degree‐balanced; a degree‐balanced Γ‐design is also balanced. The converse is not always true. We study the spectrum for orbit‐balanced, degree‐balanced, and balanced Γ‐designs of when Γ is a graph with five vertices, none of which is isolated. We also study the existence of balanced (respectively, degree‐balanced) Γ‐designs of which are not degree‐balanced (respectively, not orbit‐balanced).     

Intersection Numbers For Subspace Designs

Intersection numbers for subspace designs are introduced and q‐analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q‐analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2‐ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.     

Some New Orders of Hadamard and Skew‐Hadamard Matrices

We construct Hadamard matrices of orders and , and skew‐Hadamard matrices of orders and . As far as we know, such matrices have not been constructed previously. The constructions use the Goethals–Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.     

The Binomial k‐Clique

A finite collection C of k‐sets, where is called a k‐clique if every two k‐sets (called lines) in C have a nonempty intersection and a k‐clique is a called a maximal k‐clique if and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that , and for all . An elementary example of a maximal k‐clique is furnished by the family of all the k‐subsets of a ‐set. This k‐clique will be called the binomial k‐clique. This paper is intended to give some combinatorial characterizations of the binomial k‐clique as a maximal k‐clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k‐cliques for a fixed value of k.     

Cops and Robbers on Graphs Based on Designs

We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on Meyniel extremal families by supplying bounds on the incidence graphs of transversal designs, certain G‐designs, and BIBDs with Families of graphs with diameter 2, C₄‐free, and with unbounded chromatic number are described with the conjectured asymptotically maximum cop number.     

Four Mutually Orthogonal Latin Squares of Order 14

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012     

Computational Results for Regular Difference Systems of Sets Attaining or Being Close to the Levenshtein Bound

Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m‐subsets in a finite field of prime order to be a regular DSS for an integer m, and give a lower bound on the parameter ρ of the DSS using cyclotomic numbers. We show that when we choose ‐subsets from the multiplicative group of order e, the lower bound on ρ is independent of the choice of subsets. In addition, we present some computational results for DSSs with block sizes and , whose parameter ρ attains or comes close to the Levenshtein bound for .     

Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C‐groomings has been considered for , and completely solved for . For , it has been shown that the lower bound for the drop cost of an optimal C‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For , there are infinitely many unsettled orders; especially the case is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.     

On the number of SQSs,latin hypercubes and MDS codes

We establish that the logarithm of the number of latin d‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) orthogonal latin squares of order n is . Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is as and .     

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Triangle‐free quasi‐symmetric 2‐ designs with intersection numbers ; and are investigated. Possibility of triangle‐free quasi‐symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012