A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d1d2-1 : d1,d2D, d1 d2} contains each nonidentity element of G exactly λ times. A difference set is called abelian, nonabelian or cyclic if the underlying group is. Difference sets a.re important in design theory because they a.re equivalent to symmetric (v, k, λ) designs with a regular automorphism group. Abelian difference sets arise naturally in the solution of many problems of signal design in digital communications, including synchronization, radar, coded aperture imaging and optical image alignment. A Menon difference set (MDS) has para.meters of the form (v,k,λ) = (4N2,2N2 - N,N2 - N); alternative names used by some authors are Hadamard difference set or H-set. The Menon para.meters provide the richest source of known examples of difference sets. The central research question is: for each integer N, which groups of order 4N2 support a MDS? This question remains open, for abelian and nonabelian groups, despite a large literature spanning thirty years. The techniques so far used include algebraic number theory, character theory, representation theory, finite geometry and graph theory as well as elementary methods and computer search. Considerable progress has been made recently, both in terms of constructive and nonexistence results. Indeed some of the most surprising advances currently exist only in preprint form, so one intention of this survey is to clarify the status of the subject and to identify future research directions. Another intention is to show the interplay between the study of MDSs and several diverse branches of discrete mathematics. It is intended that a more detailed version of this survey will appear in a future publication.

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Copyright © 1993, Utilitas Mathematica Publishing. This article first appeared in Congressus Numerantium: 93 (1993), 203-207.

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