Sets of three pairwise orthogonal Steiner triple systems

AbstractTwo Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239–252) that there exist a pair of orthogonal Steiner triple systems of order v for all v≡1,3 (mod6), with v⩾7, v≠9. In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v≡1(mod6), with v⩾19 and for all v≡3(mod6), with v⩾27 with only 24 possible exceptions

Mutually Orthogonal Latin Squares

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HOPs and COPs: Frames with partitionable transversals

In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2 n . This work is motivated by the large number of applications of these designs. 1 Introduction and Definitions Let S be a set, and let fS 1 ; : : : ; S n g be a partition of S. An fS 1 ; : : : ; S n g\GammaRoom frame is an jSj \Theta jSj array, F , indexed by S, which satisfies the following properties: 1. every cell of F either is empty or contains an unordered pair of symbols of S, 2. the subarrays S i \Theta S i are empty, for 1 i n (these subarrays are referred to as holes), 3. each symbol x 62 S i occurs once in row (or column) s, for any s..

Strongly regular graphs

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