It is shown that an m x n row-latin rectangle with symbols in {1, 2, ..., k}, k greater than or equal to n, has a transversal whenever m greater than or equal to 2n-1, and that this lower bound for m is sharp. Several applications are given. One is the construction of mappings which are generalizations of complete mappings. Another is the proof of a conjecture of Dillon on the existence of difference sets in groups of order 2(2s+2) with elementary abelian normal subgroups of order 2(s+1). (C) 1998 Academic Press. [References: 14]
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