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Two constructions are described that yield an improved lower bound for the number of 2-designs with the parameters of PG( n, q), and a lower bound for the number of resolved 2-designs with the parameters of AG( n, q). [ABSTRACT FROM AUTHOR]

The parity vectors of two Latin squares of the same side n provide a necessary condition for the two squares to be biembeddable in an orientable surface. We investigate constraints on the parity vector of a Latin square resulting from structural properties of the square, and show how the parity vector of a direct product may be obtained from the parity vectors of the constituent factors. Parity vectors for Cayley tables of all Abelian groups, some non-Abelian groups, Steiner quasigroups and Steiner loops are determined. Finally, we give a lower bound on the number of main classes of Latin squares of side n that admit no self-embeddings. [ABSTRACT FROM AUTHOR]

We give a construction that produces 6-sparse Steiner triple systems of order v for all sufficiently large v of the form 3 p, p prime and p ≡ 3 (mod 4). We also give a complete list of all 429 6-sparse systems with v < 10000 produced by this construction. [ABSTRACT FROM AUTHOR]

It was shown by Gerhard Ringel that one of the three non-isomorphic Steiner triple systems of order 15 having an automorphism of order 5 may be bi-embedded as the faces of a face 2-colourable triangular embedding of K15 in a suitable orientable surface. Ringel's bi-embedding was obtained from an appropriate current graph. In a recent paper, the present authors showed that a second STS(15) of this type may also be bi-embedded. In the present paper we show that this second bi-embedding may also be obtained from a current graph. Furthermore, we exhibit a third current graph which yields a bi-embedding of the third STS(15) of this type. [ABSTRACT FROM AUTHOR]