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A Method for Constructing Quasimultiple Affine Planes of Arbitrary Order

✍ Scribed by J.F. Dillon; G.F. Stahly; M.A. Wertheimer

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 88 KB
Volume: 1
Category: Article
ISSN: 1071-5797
DOI: 10.1006/ffta.1995.1018

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✦ Synopsis

A quasimultiple affine plane of order (n) and multiplicity (\lambda) is a ((v, k, \lambda)=\left(n^{2}, n, \lambda\right)) balanced incomplete block design. For those cases where the Bruck-Ryser theorem rules out (\lambda=1) it is an interesting problem to determine the smallest actual value of (\lambda). Jungnickel [Discrete Math. 85 (1990), 177-189] has conjectured that (\lambda=\mathbf{2}) is possible for all orders. In this note we present a construction based on a result of Stahly which reduces a number of the bounds on (\lambda) given by Jungnickel. In particular, we construct quasidouble affine planes of orders 6 and 28 verifying the conjecture in these two cases. (1995 Academic Press, Inc.

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