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Exponential bounds on the number of designs with affine parameters
Author(s) -
Clark David,
Jungnickel Dieter,
Tonchev Vladimir D.
Publication year - 2010
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.20256
Subject(s) - mathematics , hyperplane , linear subspace , affine transformation , combinatorics , exponential function , discrete mathematics , combinatorial design , upper and lower bounds , geometry , mathematical analysis
It is well‐known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG ( n, q ) or AG ( n, q ), n ≥3, grows exponentially. This result was extended recently [D. Jungnickel, V. D. Tonchev, Des Codes Cryptogr, published online: 23 May, 2009] to designs having the same parameters as a projective geometry design whose blocks are the d ‐subspaces of PG ( n, q ), for any 2≤ d ≤ n −1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d ‐subspaces of AG ( n, q ), for any 2≤ d ≤ n −1, as well as resolvable designs with these parameters. An exponential lower bound is also proved for the number of non‐isomorphic resolvable 3‐designs with the same parameters as an affine geometry design whose blocks are the d ‐subspaces of AG ( n , 2), for any 2≤ d ≤ n −1. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 475–487, 2010