Nearly perfect matchings in regular simple hypergraphs

For every fixed k greater than or equal to 3 there exists a constant c(k) with the following property. Let H be a k-uniform, D-regular hypergraph on N vertices, in which no two edges contain more than one common vertex. If k > 3 then H contains a matching covering all vertices but at most c(k)ND(-1/(k-1)). If k = 3, then H contains a matching covering all vertices but at most c(3)ND(-1/2) ln(3/2) D. This improves previous estimates and implies, for example, that any Steiner Triple System on N vertices contains a matching covering all vertices but at most O(N-1/2 ln3/2 N)) improving results by various authors.