A block-size bound for Steiner 6–wise balanced designs

✍ Scribed by Michael Ira; Earl S. Kramer

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 91 KB
• Volume: 8
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(2000)8:2<141::aid-jcd6>3.0.co;2-z

✦ Synopsis

A t-wise balanced design (tBD) of type t-vY KY ! is a pair XY Bwhere X is a velement set of points and B is a collection of subsets of X called blocks with the property that the size of every block is in K and every t-element subset of X is contained in exactly ! blocks. If K is a set of positive integers strictly between t and v then we say the tBD is proper. In this paper, we presume our designs are proper. A t-vY KY ! design is also denoted by S ! tY KY v. If jKj 1, then the tBD is called a t-vY kY ! design, where K fkg. If ! 1, then the notation StY KY v is often used and the design is a Steiner system.

J. Tits in [4] proves

Lemma 1.1. Let M be the size of the smallest block in a proper Steiner tBD with v points. Then v ! M À t 1t 1X

Proof. Choose t points inside a block and a point outside that block to get a set T of size t 1 not contained in any block. Each t-subset of T determines a unique block and thus at least M À t additional points where these additional point sets are mutually disjoint.