A Generalization of Kraemer\u27s Result on Difference Sets

Kraemer has shown that every abelian group of order 22d+ 2 with exponent less than 22d+ 3 has a difference set. Generalizing this result, we show that any nonabelian group with a central subgroup of size 2d+ 1 together with an exponent-like condition will have a difference set

A Generalization of Kraemer\u27s Result on Difference Sets

Kraemer has shown that every abelian group of order 22d+ 2 with exponent less than 22d+ 3 has a difference set. Generalizing this result, we show that any nonabelian group with a central subgroup of size 2d+ 1 together with an exponent-like condition will have a difference set

A Summary of Menon Difference Sets

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,d2 ∈ D, 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

A Survey of Hadamard Difference Sets

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, d2 ∈ D, d1 ≠ d2} contains each nonidentity element of G exactly λ times. A difference set is called abelian, nonabelian or cyclic according to the properties of the underlying group. Difference sets are important in design theory because they are equivalent to symmetric (v, k, λ) designs with a regular automorphism group [L]

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (22d+2, 22d+1 ±2d, 22d±2d). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 22d+2 with exponent 2d+3 . We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case

Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t + 2 has a difference set if and only if the exponent of the group is less than or equal to 2t + 2. In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 24t + 2 with exponent 23t + 2. Thus a nonabelian 2-group G with a Hadamard difference set can have exponent |G|3/4 asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was |G|1/2 asymptotically. We use representation theory to prove that the group has a difference set

Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t + 2 has a difference set if and only if the exponent of the group is less than or equal to 2t + 2. In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 24t + 2 with exponent 23t + 2. Thus a nonabelian 2-group G with a Hadamard difference set can have exponent |G|3/4 asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was |G|1/2 asymptotically. We use representation theory to prove that the group has a difference set