Maximal Resolvable Packings and Minimal Resolvable Coverings of Triples by Quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v equivalent to 0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4,6}, {r(4), r(6)}, v) with r(6) <= 1 exists if and only if v equivalent to 0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6(2n)) for all n >= 2. (C) 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209-223, 2010