Exponential bounds on the number of designs with affine parameters |
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Authors: | David Clark Dieter Jungnickel Vladimir D Tonchev |
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Institution: | 1. Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931;2. Lehrstuhl für Diskrete Mathematik, Optimierung, und Operations Research, Universität Augsburg, D–86135 Augsburg, Germany |
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Abstract: | 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 |
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Keywords: | 2‐design 3‐design resolvable design finite affine geometry |
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