Exponential bounds on the number of designs with affine parameters

Clark, David; Jungnickel, Dieter; Tonchev, Vladimir D.

doi:10.1002/jcd.20256

Cited by 5 publications

(21 citation statements)

References 10 publications

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“…Similar results can also be obtained in the affine case: in [6] lower bounds are proved on the number of non-isomorphic designs having the same parameters as an affine geometry design AG d (n, q) whose blocks are the d-subspaces of AG(n, q), for any 2 ≤ d ≤ n − 1, as well as resolvable designs with these parameters. We note that the case d = 1 has not yet been dealt with in the affine setting.…”

Section: Boundssupporting

confidence: 59%

“…Let us simply mention one small example and then state the general corollary one obtains. For instance, applied to the parameters n = 5, d = 3, q = 2, the results of [6] imply that the number of non-isomorphic (32,8,35)-designs is larger than 10 180 , and that the number of non-isomorphic resolvable designs with these parameters is larger than 1.7 · 10 27 . Theorem 2.6.…”

Section: Boundsmentioning

confidence: 99%

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Recent results on designs with classical parameters

Jungnickel

2011

J. Geom.

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We report on recent results concerning designs with the same parameters as the classical geometric designs P G d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space P G(n, q) over the field GF (q) with q elements, where 1 ≤ d ≤ n − 1. The corresponding case of designs with the same parameters as the classical geometric designs AG d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.Mathematics Subject Classification (2010). 05B05, 51E20, 94B27.

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Section: Boundssupporting

confidence: 59%

Section: Boundsmentioning

confidence: 99%

Recent results on designs with classical parameters

Jungnickel

2011

J. Geom.

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“…It was shown in [3] that the number of non-isomorphic resolvable designs with the parameters of AG d (n, q) grows exponentially with linear growth of n whenever d ≥ 2. Analogous results were previously established for designs with the parameters of PG d (n, q); see [14] for d = 1 and [15] for d ≥ 2.…”

Section: A Lower Boundmentioning

confidence: 99%

The characterization problem for designs with the parameters of AGd(n, q)

Jungnickel

Metsch

2015

Combinatorica

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In memoriam Hanfried LenzWe start a new characterization of the geometric 2-design AG d (n, q) among all simple 2-designs with the same parameters by handling the cases d ∈ {1, 2, 3, n − 2}. For d = 1, our characterization is in terms of line sizes, and for d = 1 in terms of the number of affine hyperplanes. We also show that the number of non-isomorphic resolvable designs with the parameters of AG1(n, q) grows exponentially with linear growth of n.

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“…The constructions presented below have some similarities to constructions presented in [1,4,10,11]. In [4] and [1], new designs are created from PG d (n, q) and AG d (n, q) by rearranging parts of these designs.…”

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confidence: 96%

“…In [4] and [1], new designs are created from PG d (n, q) and AG d (n, q) by rearranging parts of these designs. In Constructions 2.1 and 2.2, we start with arbitrary constituent parts P(d , n , q) and R A(d , n , q) and combine these to form new designs P(d, n, q) and R A(d, n, q).…”

mentioning

confidence: 99%