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.

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 .

A *q*-ary code of length *n*, size *M*, and minimum distance *d* is called an code. An code with is said to be maximum distance separable (MDS). Here one-error-correcting () MDS codes are classified for small alphabets. In particular, it is shown that there are unique (5, 5^{3}, 3)_{5} and (5, 7^{3}, 3)_{7} codes and equivalence classes of (5, 8^{3}, 3)_{8} codes. The codes are equivalent to certain pairs of mutually orthogonal Latin cubes of order *q*, called Graeco-Latin cubes.

A 1-factorization of a graph *G* is a decomposition of *G* into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorization is a 1-factorization in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorizations of even order 4-regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted , which are Cayley graphs if and only if *k* is even or . By solving the perfect 1-factorization problem for a large class of graphs, we obtain a new class of 4-regular bipartite circulant graphs that do not have a perfect 1-factorization, answering a problem posed in . With further study of graphs, we prove the nonexistence of P1Fs in a class of 4-regular non-bipartite circulant graphs, which is further support for a conjecture made in .

Generalized balanced tournament packings (GBTPs) extend the concept of generalized balanced tournament designs introduced by Lamken and Vanstone (1989). In this paper, we establish the connection between GBTPs and a class of codes called equitable symbol weight codes (ESWCs). The latter were recently demonstrated to optimize the performance against narrowband noise in a general coded modulation scheme for power line communications. By constructing classes of GBTPs, we establish infinite families of optimal ESWCs with code lengths greater than alphabet size and whose narrowband noise error-correcting capability to code length ratios do not diminish to zero as the length grows.

**Every abelian group of even order with a noncyclic Sylow 2-subgroup is known to be R-sequenceable except possibly when the Sylow 2-subgroup has order 8. We construct an R-sequencing for many groups with elementary abelian Sylow 2-subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2-groups have terraces. For odd orders it is known that abelian groups are R-sequenceable except possibly those with noncyclic Sylow 3-subgroups. We show how the theory of narcissistic terraces can be exploited to find R-sequencings for many such groups, including infinitely many groups with each possible of Sylow 3-subgroup type of exponent at most** 3^{12} **and all groups whose Sylow 3-subgroups are of the form** **or** .

In alternating sign matrices, the first and last nonzero entry in each row and column is specified to be +1. Such matrices always exist. We investigate a generalization by specifying independently the sign of the first and last nonzero entry in each row and column to be either a +1 or a −1. We determine necessary and sufficient conditions for such matrices to exist whose proof contains an algorithm for their construction.

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.

Two Latin squares and , of even order *n* with entries , are said to be nearly orthogonal if the superimposition of *L* on *M* yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi-difference” sets for these orders.

A *cross-free* set of size *m* in a Steiner triple system is three pairwise disjoint *m*-element subsets such that no intersects all the three -s. We conjecture that for every admissible *n* there is an STS(*n*) with a cross-free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross-free set of size 6*k*. We note that some of the 3-bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross-free sets of size close to 6*k* (but cannot have size exactly 6*k*). The constructed STS shows that equality is possible for in the following result: in every 3-coloring of the blocks of any Steiner triple system STS(*n*) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible *n*). The analog problem can be asked for *r*-colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every *r*-coloring of the blocks of any STS(*n*), there is a monochromatic connected component with at least points, and this is sharp for infinitely many *n*.

In this paper, we consider the following question. What is the maximum number of pairwise disjoint *k*-spreads that exist in PG()? We prove that if divides and then there exist at least two disjoint *k*-spreads in PG() and there exist at least pairwise disjoint *k*-spreads in PG(*n*, 2). We also extend the known results on parallelism in a projective geometry from which the points of a given subspace were removed.

It is known that extremal ternary self-dual codes of length mod 12) yield 5-designs. Previously, mutually disjoint 5-designs were constructed by using single known generator matrix of bordered double circulant ternary self-dual codes (see [1, 2]). In this paper, a number of generator matrices of bordered double circulant extremal ternary self-dual codes are searched with the aid of computer. Using these codes we give many mutually disjoint 5-designs. As a consequence, a list of 5-spontaneous emission error designs are obtained.

]]>Let be a nontrivial 2- symmetric design admitting a flag-transitive, point-primitive automorphism group *G* of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2-(144, 66, 30) design with or , a 2-(176, 50, 14) design with , a 2-(176, 126, 90) design with or , or a 2-(14,080, 12,636, 11,340) design with .

Let *q* be an odd prime power and let be the minimum size of the symmetric difference of *r* lines in the Desarguesian projective plane . We prove some results about the function , in particular showing that there exists a constant such that for .

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1-factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order *n*, a quasigroup of order *n* or a 1-factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order *n*. For groups of order *n* it is known that automorphisms must have order less than *n*, but we show that quasigroups of order *n* can have automorphisms of order greater than *n*. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D. Stones.

A 3-uniform friendship hypergraph is a 3-uniform hypergraph in which, for all triples of vertices *x*, *y*, *z* there exists a unique vertex *w*, such that , and are edges in the hypergraph. Sós showed that such 3-uniform friendship hypergraphs on *n* vertices exist with a so-called universal friend if and only if a Steiner triple system, exists. Hartke and Vandenbussche used integer programming to search for 3-uniform friendship hypergraphs without a universal friend and found one on 8, three nonisomorphic on 16 and one on 32 vertices. So far, these five hypergraphs are the only known 3-uniform friendship hypergraphs. In this paper we construct an infinite family of 3-uniform friendship hypergraphs on 2^{k} vertices and edges. We also construct 3-uniform friendship hypergraphs on 20 and 28 vertices using a computer. Furthermore, we define *r*-uniform friendship hypergraphs and state that the existence of those with a universal friend is dependent on the existence of a Steiner system, . As a result hereof, we know infinitely many 4-uniform friendship hypergraphs with a universal friend. Finally we show how to construct a 4-uniform friendship hypergraph on 9 vertices and with no universal friend.

A uniform framework is presented for biembedding Steiner triple systems obtained from the Bose construction using a cyclic group of odd order, in both orientable and nonorientable surfaces. Within this framework, in the nonorientable case, a formula is given for the number of isomorphism classes and the particular biembedding of Ducrocq and Sterboul (preprint 18pp., 1978) is identified. In the orientable case, it is shown that the biembedding of Grannell et al. (J Combin Des **6** (), 325–336) is, up to isomorphism, the unique biembedding of its type. Automorphism groups of the biembeddings are also given.

In this article, we provide direct constructions for five mutually orthogonal Latin squares (MOLS) of orders and 60. For , these come from a new (60, 6, 1) difference matrix. For , the required construction is obtained by combining two different methods that were used in the constructions of four MOLS(14) and eight MOLS(36).

The classification of all semiovals and blocking semiovals in PG(2, 8) is determined. Also, new theoretical results on the structure of semiovals containing a -secant and some nonexistence results are presented.

In Dempwolff gives a construction of three classes of rank two semifields of order , with *q* and *n* odd, using Dembowski–Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for , in particular we prove that two of these classes, labeled and , are new for , whereas presemifields in family are isotopic to Generalized Twisted Fields for each .

The notion of a *symmetric* Hamiltonian cycle system (HCS) of a graph Γ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for , by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1–15] for , and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs and . In each case, there must be an involutory permutation ψ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for , this ψ should be precisely the permutation switching all pairs of endpoints of the edges of *I*.

An HCS is *cyclic* if it is invariant under some cyclic permutation of all the vertices. The existence question for a cyclic HCS of has been completely solved by Jordon and Morris [Discrete Math (2008), 2440–2449]—and we note that their cyclic construction is also symmetric for (mod 8). It is then natural to study the existence problem of an HCS of a graph or multigraph Γ as above which is both cyclic and symmetric. In this paper, we completely solve this problem: in the case of even order, the final answer is that cyclicity and symmetry can always cohabit when a cyclic solution exists. On the other hand, imposing that a cyclic HCS of odd order is also symmetric is very restrictive; we prove in fact that an HCS of with both properties exists if and only if is a prime.

Optical orthogonal codes (1D constant-weight OOCs or 1D CWOOCs) were first introduced by Salehi as signature sequences to facilitate multiple access in optical fibre networks. In fiber optic communications, a principal drawback of 1D CWOOCs is that large bandwidth expansion is required if a big number of codewords is needed. To overcome this problem, a two-dimensional (2D) (constant-weight) coding was introduced. Many optimal 2D CWOOCs were obtained recently. A 2D CWOOC can only support a single QoS (quality of service) class. A 2D variable-weight OOC (2D VWOOC) was introduced to meet multiple QoS requirements. A 2D VWOOC is a set of 0, 1 matrices with variable weight, good auto, and cross-correlations. Little is known on the construction of optimal 2D VWOOCs. In this paper, new upper bound on the size of a 2D VWOOC is obtained, and several new infinite classes of optimal 2D VWOOCs are obtained.