Search

The research on orthogonal Steiner systems S (t , k , v) was initiated in 1968. For (t , k) ∈ { (2 , 3) , (3 , 4) } , this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t ≥ 3 and k ≥ 5 , only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S (3 , 5 , v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S (3 , 5 , 4 m + 1) is established for any even m ≥ 2 ; additionally a pair of orthogonal G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is displayed for any odd m ≥ 3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S (5 , 8 , 24) are shown to exist. [ABSTRACT FROM AUTHOR]

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph. [ABSTRACT FROM AUTHOR]

Abstract: The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is . Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification. [Copyright &y& Elsevier]

Abstract: A Steiner triple system of order (briefly STS) consists of a -element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS (briefly LSTS) is a partition of all 3-subsets (triples) of X into STS. In 1983–1984, Lu Jiaxi first proved that there exists an LSTS for any or with six possible exceptions and a definite exception . In 1989, Teirlinck solved the existence of LSTS for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems. [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]