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A hole-size bound for incomplete t-wise balanced designs

✍ Scribed by Donald L. Kreher; Rolf S. Rees

Publisher: John Wiley and Sons
Year: 2001
Tongue: English
Weight: 140 KB
Volume: 9
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.1011

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✦ Synopsis

Abstract

An incomplete t‐wise balanced design of index λ is a triple (X,H,ℬ︁) where X is a υ–element set, H is a subset of X called the hole, and B is a collection of subsets of X called blocks, such that, every t‐element subset of X is either in H or in exactly λ blocks, but not both. If H is a hole in an incomplete t‐wise balanced design of order υ and index λ, then |H| ≤ υ/2 if t is odd and |H| ≤ (υ − 1)/2 if t is even. In particular, this result establishes the validity of Kramer's conjecture that the maximal size of a block in a Steiner t‐wise balanced design is at most υ/2 if t is odd and at most (υ−1)/2 when t is even. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 269–284, 2001

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## Abstract 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, They showed that when __t__ = 2 or 3, this bound is sharp infinitely often in that for each __h__ ≥ __t__ and each __k__ ≥ _