Further results on the maximum size of a hole in an incomplete t‐wise balanced design with specified minimum block size *

Kreher and Rees 3 proved that if h is the size of a hole in an incomplete balanced design of order υ and index λ having minimum block size $k \ge t+1$ , then,$$h \leq {{v}+(k-t)(t-2)-1\over k-t+1}.$$ They showed that when t  = 2 or 3, this bound is sharp infinitely often in that for each h  ≥  t and each k  ≥  t  + 1, ( t,h,k ) ≠(3,3,4), there exists an I t BD meeting the bound. In this article, we show that this bound is sharp infinitely often for every t , viz., for each t  ≥ 4 there exists a constant C t  > 0 such that whenever ( h − t )( k − t  − 1) ≥  C t there exists an I t BD meeting the bound for some λ = λ( t,h,k ). We then describe an algorithm by which it appears that one can obtain a reasonable upper bound on C t for any given value of t . © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 256–281, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10014