The number of designs with classical parameters grows exponentially
type: Item
instance of: scholarly article
title: The number of designs with classical parameters grows exponentially
zbMATH Open document ID: 0546.05008
DOI: 10.1007/BF00146828
author: Dieter Jungnickel
published in: Geometriae Dedicata
publication date: 1984-01-01T00:00:00Z
review text: The number of non-isomorphic block designs S(2,k;v) grows very fast with v. \textit{R. M. Wilson} [Math. Z. 135, 303-313 (1974; Zbl 0269.05005)] proved that it is at least exp \(\alpha\) v\({}^ 2\) with \(\alpha >0\) for large v. For affine and symmetric designs other methods are necessary. The author shows that the number of non-isomorphic affine designs with the parameters of affine spaces \(AG_{d-1}(d,q)\) [the blocks are the hyperplanes] grows exponentially with q. The same holds for symmetric designs with the parameters of projective spaces \(PG_{d-1}(d,q)\). Similar estimates hold for biaffine divisible designs and symmetric nets (for details see the paper). In the formula on p. 170, row 3 from below, read \((q-1)^ 2\) instead of \((q-q)^ 2\).
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 51E05
zbMATH DE Number: 3869337
zbMATH Keywords: number of non-isomorphic block designs
zbMATH Keywords: affine designs
zbMATH Keywords: divisible designs
reviewed by: Hanfried Lenz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00146828
OpenAlex ID: W2014666954
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description: scientific article; zbMATH DE number 3869337