There is no Drake-Larson finite linear space with three 7-lines and twenty-four 5-lines
✍ Scribed by Melissa Berg; Ronald Mullin
- Publisher
- John Wiley and Sons
- Year
- 2008
- Tongue
- English
- Weight
- 138 KB
- Volume
- 16
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
Let L denote the set of positive integers that do not divide 6. In 1983, Drake and Larson determined all positive integers v such that there is a proper PBD(v,L) with one exception, namely, the integer v = 30. (Recall that a PBD is proper if there is no block that contains all of the points and every block contains at least two points of the design.) In 1984, they showed that in any proper PBD(30, L), there could only be blocks of sizes in {4, 5, 7, 8}, and if b~i~ denotes the number of blocks of size i in such a design, then there are only 6 possible block distributions (b~8~,b~7~,b~5~,b~4~), namely, (1,1,14,41), (0,3,24,22 ), (0,3,15,37), (0,1,27,24), (0,1,24,29), and (0,1,15,44). In 2004, Grüttmüller and Streso eliminated the first of these. In the current article we use the geometry of a putative proper PBD with block distribution (0,3,24,22) to eliminate this case. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 191–201, 2008