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Lander’s tables are complete! (English) Zbl 0940.05016
Pott, A. (ed.) et al., Difference sets, sequences and their correlation properties. Proceedings of the NATO Advanced Study Institute, Bad Winsheim, Germany, August 2-14, 1998. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 542, 239-257 (1999).
Summary: In 1983 E. S. Lander published “Symmetric designs: An algebraic approach”; see [Lond. Math. Soc. Lect. Note Ser. 74 (1983; Zbl 0502.05010)]. At the back are provided a set of tables. Each entry in a table specifies an abelian group G, the parameters of a putative difference set D in G with cardinality k, whether D exists or not, a construction when D does exist, and a nonexistence proof when D does not exist. At time of publication there were some 25 entries which were open, i.e. the existence or nonexistence of D had not been determined. From 1983 until 1993 K. T. Arasu and others filled in all but 8 entries. In 1994 entry 148 was filled in by J. E. Iiams, R. A. Liebler and K. W. Smith. In this paper we fill in the last seven entries by demonstrating nonexistence. We point out several applications to nonabelian difference sets.
MSC:
05B10Difference sets