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On abelian difference sets. (English) Zbl 0927.05011
Passi, I. B. S. (ed.), Algebra. Some recent advances. Basel: Birkhäuser. Trends in Mathematics. 1-27 (1999).
The paper is a short survey on abelian difference sets. It contains a description of the known series. The most important and classical concepts in the theory (multipliers, characters, tools from algebraic number theory) are explained. The paper contains no proofs, the results are just summarized. It is mostly intended for an impatient reader who wants to know only basic facts about difference sets. Important recent developments are neglected, mostly due to the fact that 27 pages are not enough to cover all the exciting new results on difference sets discovered recently. For instance, the concept of building sets due to Davis and Jedwab is treated only in a few sentences; see J. Davis and J. Jedwab [A unifying construction for difference sets, J. Comb. Theory, Ser. A 80, No. 1, 13-78 (1997; Zbl 0884.05019)]. Basically nothing is said about Schmidt’s important new nonexistence results; see B. Schmidt [Cyclotomic integers and finite geometry, J. Am. Math. Soc. 12, No. 4, 929-952 (1999)]. The paper contains an interesting list of the existence status of abelian difference sets with $k\le 150$.
##### MSC:
 05B10 Difference sets 05B05 Block designs (combinatorics) 05B25 Finite geometries (combinatorics) 05B20 Matrices (incidence, Hadamard, etc.)