A Unifying Construction for Difference Sets

Authors

James A. Davis, University of RichmondFollow
Jonathan Jedwab

DOI

10.1006/jcta.1997.2796

Abstract

We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, λ,n)=(2^2d+4(2^2d+2−1)/3, 2^2d+1(2^2d+3+1)/3, 2^2d+1(2^2d+1+1)/3, 2^4d+2) for d⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 2^2d+3(2^2d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order p^r. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups.

Document Type

Article

Publication Date

10-1997

Publisher Statement

Copyright © 1997, Elsevier. This article first appeared in Journal of Combinatorial Theory, Series A: 80:1 (1997), 13-78.

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Recommended Citation

Davis, James A., and Jonathan Jedwab. "A Unifying Construction for Difference Sets." Journal of Combinatorial Theory, Series A 80, no. 1 (October 1997): 13-78. doi: 10.1006/jcta.1997.2796.