173 research outputs found
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 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 Unified Approach to Difference Sets with gcd(\u3ci\u3eV, N\u3c/i\u3e) \u3e 1
The five known families of difference sets whose parameters (v, k, λ; n) satisfy the condition gcd(v,n) \u3e 1 are the McFarland, Spence, Davis-Jedwab, Hadamard and Chen families. We survey recent work which uses recursive techniques to unify these difference set families, placing particular emphasis on examples. This unified approach has also proved useful for studying semi-regular relative difference sets and for constructing new symmetric designs
A Unifying Construction for Difference Sets
We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, λ,n)=(22d+4(22d+2−1)/3, 22d+1(22d+3+1)/3, 22d+1(22d+1+1)/3, 24d+2) for d⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order pr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r\u3e1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups
Subsets of finite groups exhibiting additive regularity
In this article we aim to develop from first principles a theory of sum sets
and partial sum sets, which are defined analogously to difference sets and
partial difference sets. We obtain non-existence results and characterisations.
In particular, we show that any sum set must exhibit higher-order regularity
and that an abelian sum set is necessarily a reversible difference set. We next
develop several general construction techniques under the hypothesis that the
over-riding group contains a normal subgroup of order 2. Finally, by exploiting
properties of dihedral groups and Frobenius groups, several infinite classes of
sum sets and partial sum sets are introduced
Constructions of difference sets in nonabelian 2-groups
Difference sets have been studied for more than 80 years. Techniques from
algebraic number theory, group theory, finite geometry, and digital
communications engineering have been used to establish constructive and
nonexistence results. We provide a new theoretical approach which dramatically
expands the class of -groups known to contain a difference set, by refining
the concept of covering extended building sets introduced by Davis and Jedwab
in 1997. We then describe how product constructions and other methods can be
used to construct difference sets in some of the remaining -groups. We
announce the completion of ten years of collaborative work to determine
precisely which of the 56,092 nonisomorphic groups of order 256 contain a
difference set. All groups of order 256 not excluded by the two classical
nonexistence criteria are found to contain a difference set, in agreement with
previous findings for groups of order 4, 16, and 64. We provide suggestions for
how the existence question for difference sets in -groups of all orders
might be resolved.Comment: 38 page
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
(2^n,2^n,2^n,1)-relative difference sets and their representations
We show that every -relative difference set in
relative to can be represented by a polynomial f(x)\in \F_{2^n}[x],
where is a permutation for each nonzero . We call such an
a planar function on \F_{2^n}. The projective plane obtained from
in the way of Ganley and Spence \cite{ganley_relative_1975} is
coordinatized, and we obtain necessary and sufficient conditions of to be
a presemifield plane. We also prove that a function on \F_{2^n} with
exactly two elements in its image set and is planar, if and only if,
for any x,y\in\F_{2^n}
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