A New Bound on the Number of Designs with Classical Affine Parameters

Authors: Lam C.¹; Tonchev V.D.²

Source: Designs, Codes and Cryptography, Volume 27, Numbers 1-2, October 2002 , pp. 111-117(7)

Publisher: Springer

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Abstract:

The hyperplanes in the affine geometry AG(d, q) yield an affine resolvable design with parameters 2-(q^d, q^{d-1}, \frac{q^{d-1} -1}{q-1}). Jungnickel [3]proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parameters for d 3. The bound of Jungnickel was improved recently [5] by a factor of q^{\frac{d^2+d-6}{2}}(q-1)^{d-2} for any d 4. In this paper, a construction of 2-(q^d, q^{d-1}, \frac{q^{d-1}-1}{q-1}) designs based on group divisible designs is given that yields at least \frac{(q^{d-1}+q^{d-2}+\cdots+1)!(q-1)}{|{\rm P}\Gamma {\rm L}(d,q)||{\rm A}\Gamma {\rm L}(d,q)|} non-isomorphic designs for any d 3. This new bound improves the bound of [5] by a factor of \frac{1}{q^d}\prod_{i=1}^{(q^{d-1}-q)/(q-1)}(q^{d-1}+i).For any given q and d, It was previously known [2,11] that there are at least 8071 non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is at least 245,100,000.

Language: English

Document Type: Research article

Affiliations: 1: Department of Computer Science, Concordia University, Montreal, Quebec H3G 1M8, Canada 2: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA

Publication date: 2002-10-01