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Journal of the London Mathematical Society 1986 s2-34(2):193-211; doi:10.1112/jlms/s2-34.2.193
© 1986 by London Mathematical Society
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© Oxford University Press

Irregular Extremal Perfect Systems of Difference Sets

D. G. Rogers

68 Liverpool Road, Watford WD1 8DN

An (m, n;u, v;c)-system is a perfect system of difference sets with m components of size u – 1 and n components of size v – 1 having threshold c. A necessary condition for the existence of an (m, n; u, 6; c)-system for u = 3 or 4 is that m ≥ 2c – 1. We show that, in the extremal case when m = 2c –1, there are (2c – 1, n; 3, 6; c)-systems for c ≥ 3 when n = 1, and for c ≥ 5 when n = 2, with a few possible exceptions, by first characterizing the components of size 5 and then drawing on results on complete permutations with constraints to construct the components of size 3. Large classes of non-extremal (m, n; 3, 6; c)-systems can be constructed from these extremal families; and the thrust of our work is to suggest that the necessary condition that m ≥ 2c – 1 is also sufficient when c is sufficiently large compared with n.

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