A connected covering is a design system in which the corresponding *block graph* is connected. The minimum size of such coverings are called *connected coverings numbers*. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with *Turán systems*. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.

An is a triple , where *X* is a set of points, is a partition of *X* into *m* disjoint sets of size *n* and is a set of 4-element transverses of , such that each 3-element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of *n* cycles of length *m* (resp. *m* cycles of length *n*), then this is called *m*-cyclic (resp. semi-cyclic). Further, if all block-orbits of an *m*-cyclic (resp. semi-cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly *m*-cyclic and semi-cyclic , and use them to give new infinite classes of perfect two-dimensional optical orthogonal codes with maximum collision parameter and AM-OPPTS/AM-OPPW property.

A cube design of order *v*, or a CUBE(*v*), is a decomposition of all cyclicly oriented quadruples of a *v*-set into oriented cubes. A CUBE(*v*) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate.

We show that a nondegenerate CUBE(*v*) design exists for all integers , and that an unoriented nondegenerate CUBE(*v*) design exists if and only if and or . A degenerate example of a CUBE(*v*) design is also given for each integer .

Using the Katona–Kierstead (K–K) definition of a Hamilton cycle in a uniform hypergraph, we investigate the existence of wrapped K–K Hamilton cycle decompositions of the complete bipartite 3-uniform hypergraph and their large sets, settling their existence whenever *n* is prime.

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with *m* odd, where denotes the first kind of Dickson polynomials of order *n* and . The key observation in the proof is that is a planar function from to for *m* odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where *m* is odd and . The proof is more complicated and different than that of Ding-Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.

We detail the enumeration of all two-intersection sets of the five-dimensional projective space over the field of order 3 that are invariant under an element of order 7, which include the examples of Hill (1973) and Gulliver (1996). Up to projective equivalence, there are 6,635 such two-intersection sets.

Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1-factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .

We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.

Let *L* be a latin square of indeterminates. We explore the determinant and permanent of *L* and show that a number of properties of *L* can be recovered from the polynomials and per(*L*). For example, it is possible to tell how many transversals *L* has from per(*L*), and the number of 2 × 2 latin subsquares in *L* can be determined from both and per(*L*). More generally, we can diagnose from or per(*L*) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in and per(*L*) that involves only two different indeterminates. Latin squares *A* and *B* are *trisotopic* if *B* can be obtained from *A* by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic latin squares with equal permanents and equal determinants exist for all orders that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.

We consider the existence problem for a semi-cyclic holey group divisible design of type with block size 3, which is denoted by a 3-SCHGDD of type . When *t* is odd and or *t* is doubly even and , the existence problem is completely solved; when *t* is singly even, many infinite families are obtained. Applications of our results to two-dimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

If a cycle decomposition of a graph *G* admits two resolutions, and , such that for each resolution class and , then the resolutions and are said to be *orthogonal*. In this paper, we introduce the notion of an orthogonally resolvable cycle decomposition, which is a cycle decomposition admitting a pair of orthogonal resolutions. An orthogonally resolvable cycle decomposition of a graph *G* may be represented by a square array in which each cell is either empty or filled with a *k*–cycle from *G*, such that every vertex appears exactly once in each row and column of the array and every edge of *G* appears in exactly one cycle. We focus mainly on orthogonal *k*-cycle decompositions of and (the complete graph with the edges of a 1-factor removed), denoted . We give general constructions for such decompositions, which we use to construct several infinite families. We find necessary and sufficient conditions for the existence of an OCD(*n*, 4). In addition, we consider orthogonal cycle decompositions of the lexicographic product of a complete graph or cycle with . Finally, we give some nonexistence results.

A *pseudo-hyperoval* of a projective space , *q* even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that is flag-transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).

We provide general criteria for orthogonal arrays and t-designs on equitable partitions of a hypercube by exploring harmonic distributions. Generalized harmonic weight enumerators for real-valued functions of are introduced and applied to eigenfunctions of the adjacency matrix of . Using this, expressions for harmonic distributions are established for every cell of an equitable partition π of . Moreover, for any given cell in the partition π, the strength of the cell as an orthogonal array is explicitly expressed, and also a characterization of a t-design of that cell is established. We also compute strengths of cells and find t-designs from cells based on constructions of Krotov, Borges, Rifa, and Zinoviev.

We obtained a full computer classification of all complete arcs in the Desarguesian projective plane of order 31 using essentially the same methods as for earlier results for planes of smaller order, i.e., isomorph-free backtracking using canonical augmentation. We tabulate the resulting numbers of complete arcs according to size and automorphism group. We give explicit descriptions for all complete arcs with an automorphism group of size at least 20. In some of these cases the constructions can be generalized to other values of *q*. In particular, we find arcs of size 20 for any field of order , and a complete 44-arc in PG(2,67) with an automorphism group of order 88. We also correct a result by Kéri : there are 12 complete 22-arcs in PG(2,31) up to projective equivalence, and not 11.

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1-dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares

Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any , there exists a CQS for all , and .

Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.

A -semiframe of type is a -GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A -SF is a -semiframe of type in which there are *p* parallel classes in and *d* holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)-SF for any if and only if , , , and .

2-(*v,k,1*) designs admitting a primitive rank 3 automorphism group , where *G*_{0} belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in [23], are classified.

A *G*-design of order *n* is a decomposition of the complete graph on *n* vertices into edge-disjoint subgraphs isomorphic to *G*. Grooming uniform all-to-all traffic in optical ring networks with grooming ratio *C* requires the determination of graph decompositions of the complete graph on *n* vertices 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 existence spectrum problem of *G*-designs for five-vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such *G*-designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five-vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8-groomings for all orders *n* by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9-groomings for all orders *n* is realizable with 14 exceptions and 12 possible exceptions.

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 .

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.

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*.

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.

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.

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.

The constructed pandiagonal Latin squares by Hedayat's method are cyclic. During the last decades several authors described methods for constructing pandiagonal Latin squares that are semi-cyclic. In this article, we have applied linear cellular automaton on elements of permutation elementary abelian *p*-groups, which are ordered by the lexicographic ordering, and we proposed an algorithm for constructing noncyclic pandiagonal Latin squares of order , for prime .