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An H(m,g,k,3) design is a triple (X,T,B), where X is a set of mg points, T a partition of X into m disjoint sets of size g, and B a set of k‐element transverses of T, such that each 3‐element transverse of T is contained in exactly one of them. In 1990, Mills determined the existence of an H(m,g,4,3) design with m≠5. In this paper, an efficient construction shows that an H(5,g,4,3) exists for any integer g≡2,10(mod12) with g≥10. Consequently, the necessary and sufficient conditions for the existence of an H(m,g,4,3) design are m≥4, mg≡0(mod2), and g(m−1)(m−2)(mod3), with a definite exception (m,g)=(5,2). [ABSTRACT FROM AUTHOR]

A dihedral SQS is an SQS admitting a point-regular dihedral automorphism group. Some constructions of dihedral SQSs are established in this paper. We also prove that for v ≡ 2 , 4 ( m o d 6 ) , there is a dihedral SQS ( v ) if there is a dihedral H ( p , 2 , 4 , 3 ) design or an S -cyclic SQS ( 2 p ) for each odd prime divisor p of v . As a corollary, we enlarge the set of known values of v for which there exists a dihedral SQS ( v ) . [ABSTRACT FROM AUTHOR]

Abstract: A Steiner system is a pair , where is a -element set and is a set of -subsets of , called blocks, with the property that every -element subset of is contained in a unique block. The sub-design in a Steiner quadruple system is said to be a spanning block design. The order of a Steiner quadruple system with a spanning block design should satisfy the necessary condition . It is proved that the above necessary condition is also sufficient. As a consequence, it is also proved that a 3-BD exists for any . [Copyright &y& Elsevier]

Abstract: The arboricity of a hypergraph is the minimum number of acyclic hypergraphs that partition . The determination of the arboricity of hypergraphs is a problem motivated by database theory. The exact arboricity of the complete -uniform hypergraph of order is previously known only for . The arboricity of the complete -uniform hypergraph of order is determined asymptotically when , positive, and determined exactly when . This proves a conjecture of Wang (2008)  in the asymptotic sense. [Copyright &y& Elsevier]

Abstract: An augmented Steiner quadruple system of order is an ordered triple , where is an SQS and is the set of all 2-subsets of . An augmented Steiner quadruple system of order is resolvable if can be partitioned into parts such that each part is a partition of . Hartman and Phelps in [A. Hartman, K.T. Phelps, Steiner quadruple systems, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory, Wiley, New York, 1992, pp. 205–240] conjectured that there exists a resolvable augmented Steiner quadruple systems of order for any positive integer or 10 (mod 12). In this paper, we show that the Hartman and Phelps conjecture is true. [Copyright &y& Elsevier]

Abstract: A -wise balanced design is said to be resolvable if its block set can be partitioned into parts (called resolution classes) such that each part is itself a partition of the point set. It is uniform if all blocks in each resolution class have the same size. In this paper, it is shown that a uniformly resolvable three-wise balanced design of order with block sizes four and six exists if and only if is divisible by 4. These uniformly resolvable three-wise balanced designs are also used to construct the infinite classes of resolvable maximal packings (minimal coverings) of triples by quadruples of order for , augmented resolvable Steiner quadruple systems of order for and -resolvable Steiner quadruple systems of order for . [Copyright &y& Elsevier]

Abstract: In this paper we give some recursive constructions for strictly cyclic 3-designs. Using these constructions we have some infinite families of strictly cyclic Steiner quadruple systems and optimal optical orthogonal codes with weight 4 and index 2. As corollaries, many known constructions for strictly cyclic Steiner quadruple systems and optimal optical orthogonal codes are unified. We also notice that there does not exist an optimal -OOC for any (). Thus we introduce the concept of strictly cyclic maximal packing quadruple systems to deal with the cases of () for -OOCs. By our recursive constructions, some infinite families are also given on strictly cyclic maximal packing quadruple systems. [Copyright &y& Elsevier]