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.

This paper is intended as a first step toward a general Sylow theory for quasigroups and Latin squares. A subset of a quasigroup lies in a *nonoverlapping orbit* if its respective translates under the elements of the left multiplication group remain disjoint. In the group case, each nonoverlapping orbit contains a subgroup, and Sylow's Theorem guarantees nonoverlapping orbits on subsets whose order is a prime-power divisor of the group order. For the general quasigroup case, the paper investigates the relationship between non-overlapping orbits and structural properties of a quasigroup. Divisors of the order of a finite quasigroup are classified by the behavior of nonoverlapping orbits. In a dual direction, Sylow properties of a subquasigroup *P* of a finite left quasigroup *Q* may be defined directly in terms of the homogeneous space , and also in terms of the behavior of the isomorphism type within the so-called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.

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 .

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.

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.

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 .

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).

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function *T* such that no 3-phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size exists, then .

In [8] 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 .

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.

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