A complete solution to existence of H designs. Issue 2 (13th November 2018) - Ji, Lijun
Abstract: An H ( m, g, k, 3 ) design is a triple ( X, T, B ), where X is a set of m g points, T a partition of X into m disjoint sets of size g, and B a set of k ‐element transverses of T, such that each 3 ‐element transverse of T is contained in exactly one of them. In 1990, Mills determined the existence of an H ( m, g, 4, 3 ) design with m ≠ 5 . In this paper, an efficient construction shows that an H ( 5, g, 4, 3 ) exists for any integer g ≡ 2, 10 ( mod 12 ) with g ≥ 10 . Consequently, the necessary and sufficient conditions for the existence of an H ( m, g, 4, 3 ) design are m ≥ 4, m g ≡ 0 ( mod 2 ), and g ( m − 1 ) ( m − 2 ) ( mod 3 ), with a definite exception ( m, g ) = ( 5, 2 ) .