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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]

Abstract: The problem of the existence of large sets of Kirkman triple systems (LKTS) is one of the most celebrated open problems in design theory. Only a few sparsely distributed infinite classes have been determined, although LKTS have been investigated by many authors. The purpose of this paper is to survey constructions and results on LKTS and related designs. A systematic account of this work is provided. Most of the known constructions are unified and generalized; the approaches to LKTS are enriched; a couple of new constructions for related designs are also displayed. In particular, a new existence class of LKTS is demonstrated and some new results of overlarge sets of Kirkman triple systems are also produced. [Copyright &y& Elsevier]