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Small quasimultiple affine and projective planes: Some improved bounds
Author(s) -
Buratti Marco
Publication year - 1998
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/(sici)1520-6610(1998)6:5<337::aid-jcd3>3.0.co;2-g
Subject(s) - mathematics , combinatorics , prime (order theory) , projective plane , multiplicative function , prime power , affine transformation , order (exchange) , integer (computer science) , bounded function , discrete mathematics , pure mathematics , mathematical analysis , geometry , finance , computer science , economics , correlation , programming language
We improve the known bounds on r ( n ): = min {λ| an ( n 2 , n , λ)‐RBIBD exists} in the case where n + 1 is a prime power. In such a case r ( n ) is proved to be at most n + 1. If, in addition, n − 1 is the product of twin prime powers, then r ( n ) ${\ \le \ }{n \over 2}$. We also improve the known bounds on p ( n ): = min{λ| an ( n 2 + n + 1, n + 1, λ)‐BIBD exists} in the case where n 2 + n + 1 is a prime power. In such a case p ( n ) is bounded at worst by $\lfloor {n+1 \over 2} \rfloor$ but better bounds could be obtained exploiting the multiplicative structure of GF( n 2 + n + 1). Finally, we present an unpublished construction by M. Greig giving a quasidouble affine plane of order n for every positive integer n such that n − 1 and n + 1 are prime powers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 337–345, 1998