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Further results on the maximum size of a hole in an incomplete t‐wise balanced design with specified minimum block size *
Author(s) -
Adamczak I.,
Kreher D. L.,
Ling A. C. H.,
Rees R. S.
Publication year - 2002
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.10014
Subject(s) - combinatorics , mathematics , upper and lower bounds , block (permutation group theory) , block size , order (exchange) , discrete mathematics , mathematical analysis , computer science , computer security , finance , key (lock) , economics
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