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 .

Let *n*, *k*, and *t* be integers satisfying . A Steiner system with parameters *t*, *k*, and *n* is a *k*-uniform hypergraph on *n* vertices in which every set of *t* distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for . In this note we prove that for every and sufficiently large *n*, there exists an almost Steiner system with parameters *t*, *k*, and *n*; that is, there exists a *k*-uniform hypergraph on *n* vertices such that every set of *t* distinct vertices is covered by either one or two edges.

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 .

For two graphs *G* and *H* their *wreath product* has vertex set in which two vertices and are adjacent whenever or and . Clearly, , where is an independent set on *n* vertices, is isomorphic to the complete *m*-partite graph in which each partite set has exactly *n* vertices. A 2-regular subgraph of the complete multipartite graph containing vertices of all but one partite set is called *partial* 2*-factor*. For an integer λ, denotes a graph *G* with uniform edge multiplicity λ. Let *J* be a set of integers. If can be partitioned into edge-disjoint partial 2-factors consisting cycles of lengths from *J*, then we say that *has a* *-cycle frame*. In this paper, we show that for and , there exists a -cycle frame of if and only if and . In fact our results completely solve the existence of a -cycle frame of .

In this paper, two related problems are completely solved, extending two classic results by Colbourn and Rosa. In any partial triple system of , the neighborhood of a vertex *v* is the subgraph induced by . For (mod 3) with , it is shown that for any 2-factor *F* on or vertices, there exists a maximum packing of with triples such that *F* is the neighborhood of some vertex if and only if , thus extending the corresponding result for the case where or 1 (mod 3) by Colbourn and Rosa. This result, along with the companion result of Colbourn and Rosa, leads to a complete characterization of quadratic leaves of λ-fold partial triple systems for all , thereby extending the solution where by Colbourn and Rosa.

A *generalized hyperfocused arc* in is an arc of size *k* with the property that the secants can be blocked by a set of points not belonging to the arc. We show that if *q* is a prime and is a generalized hyperfocused arc of size *k*, then or 4. Interestingly, this problem is also related to the (strong) cylinder conjecture ( Problem 919), as we point out in the last section.

A Steiner system is called a Steiner quintuple systems of order *v*. The smallest order for which the existence, or otherwise, of a Steiner quintuple system is unknown is 21. In this article, we prove that, if an *S*(4, 5, 21) exists, the order of its full automorphism group is 1, 2, 3, 4, 5, 6, 7, or 10.

Let *q* be an odd prime power such that *q* is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group *G* isomorphic to the alternating group *A*_{5}. Transitive *G*-invariant 30-arcs are shown to exist for every . The completeness is also investigated, and complete 30-arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .

New families of complete caps in finite Galois spaces are obtained. For most pairs with and , they turn out to be the smallest known complete caps in . Our constructions rely on the bicovering properties of certain plane arcs contained in plane cubic curves with a cusp.

In this paper, we are concerned about optimal (*v*, 4, 3, 2)-OOCs. A tight upper bound on the exact number of codewords of optimal (*v*, 4, 3, 2)-OOCs and some direct and recursive constructions of optimal (*v*, 4, 3, 2)-OOCs are given. As a result, the exact number of codewords of an optimal (*v*, 4, 3, 2)-OOC is determined for some infinite series.

It was shown by Babai in 1980 that almost all Steiner triple systems are rigid; that is, their only automorphism is the identity permutation. Those Steiner triple systems with the largest automorphism groups are the projective systems of orders . In this paper, we show that each such projective system may be transformed to a rigid Steiner triple system by at most *n* Pasch trades whenever .

A graph *G* of order *n* is called *t-edge-balanced* if *G* satisfies the property that there exists a positive λ for which every graph of order *n* and size *t* is contained in exactly λ distinct subgraphs of isomorphic to *G*. We call λ the *index* of *G*. In this article, we obtain new infinite families of 2-edge-balanced graphs.

A *q*-packing design is a selection of *k*-dimensional subspaces of such that each *t*-dimensional subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer–Mesner method of prescribing a group of automorphisms was applied by Kohnert and Kurz to construct some constant dimension codes with moderate parameters that arise by *q*-packing designs. In this paper, we recall this approach and give a version of the Kramer–Mesner method breaking the condition that the whole *q*-packing design must admit the prescribed group of automorphisms. Afterwards, we describe the basic idea of an algorithm to tackle the integer linear optimization problems representing the *q*-packing design construction by means of a metaheuristic approach. Finally, we give some improvements on the size of *q*-packing designs.