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There is no Drake‐Larson finite linear space with three 7‐lines and twenty‐four 5‐lines
Author(s) -
Berg Melissa,
Mullin Ronald
Publication year - 2008
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.20177
Subject(s) - mathematics , combinatorics , block (permutation group theory) , integer (computer science) , block size , space (punctuation) , distribution (mathematics) , set (abstract data type) , discrete mathematics , computer science , mathematical analysis , computer security , key (lock) , programming language , operating system
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