Equitable resolvable coverings

In an earlier article, Willem H. Haemers has determined the minimum number of parallel classes in a resolvable 2‐( qk , k ,1) covering for all k  ≥ 2 and q  = 2 or 3. Here, we complete the case q  = 4, by construction of the desired coverings using the method of simulated annealing. Secondly, we look at equitable resolvable 2‐( qk , k ,1) coverings. These are resolvable coverings which have the additional property that every pair of points is covered at most twice. We show that these coverings satisfy k < 2q −  $\sqrt{2q - {9\over4}}$ , and we give several examples. In one of these examples, k  >  q . © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 113–123, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10024