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 ≡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 ≡0 ( mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6 2 n ) for all n ≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010