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In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2‐chromatic Steiner quadruple system of order v $v$ (SQS(v) $(v)$) for all v≡4 $v\equiv 4$ or 8(mod12) $8\,(\mathrm{mod}\,12)$. The first author presented a construction for 2‐chromatic SQSs based on 2‐chromatic candelabra quadruple systems and s $s$‐fan designs. In this paper, it is proved that a 2‐chromatic SQS(v) $(v)$ also exists if v≡10(mod12) $v\equiv 10\,(\mathrm{mod}\,12)$, or if v≡2(mod24) $v\equiv 2\,(\mathrm{mod}\,24)$. [ABSTRACT FROM AUTHOR]

A t- (v , k , λ) design is a pair (X , B) , where X is a v-element set and B is a set of k-subsets of X, called blocks, with the property that every t-subset of X is contained in exactly λ blocks. A t- (v , k , λ) design (X , B) is said to be (s , μ) -resolvable if B can be partitioned into B 1 | ⋯ | B c such that each (X , B i) is an s- (v , k , μ) design, further, if each (X , B i) is also (r , ν) -resolvable, then such an (s , μ) -resolvable t-design is called (s , μ) (r , ν) -doubly resolvable. In 1980, Hartman constructed a (2, 3)(1, 1)-doubly resolvable 3-(v, 4, 1) design for v ∈ { 20 , 32 , 44 , 68 , 80 , 104 } and a (2, 3)-resolvable 3- (2 7 , 4 , 1) design. In this paper, we construct (2, 3)(1, 1)-doubly resolvable 3- (2 2 n + 1 , 4 , 1) designs for all positive integers n. [ABSTRACT FROM AUTHOR]

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]

Let [ABSTRACT FROM AUTHOR]

Abstract: Franek et al. recently described four constructions for large sets of -intersecting Steiner triple systems of order (STS ) [F. Franek, M.J. Grannell, T.S. Griggs, A. Rosa, On the large sets of -intersecting Steiner triple systems of order , Des. Codes Cryptogr. 26 (2002) 243–256]. In this study we focus on large sets of -intersecting STS . Some recursive constructions are presented that involve three-wise balanced designs. By applying these constructions we obtain some new infinite classes of such large sets, and the large sets of -intersecting KTS are used to produce some new large sets of Kirkman triple systems. [Copyright &y& Elsevier]

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: 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]

Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS (v)) (K , B) is called symmetric K -invariant if for each B ∈ B , it holds that B + x ∈ B for each x ∈ K and B = − B + y for some y ∈ K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K -invariant SQS (v) was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic SQS (v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K -invariant SQS (v) exists if and only if v ≡ 2 , 4 (mod 6) , the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS (2 p) for any odd prime divisor p of v. [ABSTRACT FROM AUTHOR]