9,680 research outputs found
Supplementary difference sets with symmetry for Hadamard matrices
First we give an overview of the known supplementary difference sets (SDS)
(A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is
either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson
matrices over the elementary abelian groups of order 25, 27 and 49 are
constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127
are presented. The last of these is obtained from a (127,57,76)-difference
family that we have constructed. An old non-published example of G-matrices of
order 37 is also included.Comment: 16 pages, 2 tables. A few minor changes are made. The paper will
appear in Operators and Matrice
On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs
In this paper, we make specific conjectures about the distribution of
Jacobians of random graphs with their canonical duality pairings. Our
conjectures are based on a Cohen-Lenstra type heuristic saying that a finite
abelian group with duality pairing appears with frequency inversely
proportional to the size of the group times the size of the group of
automorphisms that preserve the pairing. We conjecture that the Jacobian of a
random graph is cyclic with probability a little over .7935. We determine the
values of several other statistics on Jacobians of random graphs that would
follow from our conjectures. In support of the conjectures, we prove that
random symmetric matrices over the p-adic integers, distributed according to
Haar measure, have cokernels distributed according to the above heuristic. We
also give experimental evidence in support of our conjectures.Comment: 20 pages. v2: Improved exposition and appended code used to generate
experimental evidence after the \end{document} line in the source file. To
appear in J. Algebraic Combi
Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices
In this paper we bring together results about the density of subsemigroups of
abelian Lie groups, the minimal number of topological generators of abelian Lie
groups and a result about actions of algebraic groups. We find the minimal
number of generators of a finitely generated abelian semigroup or group of
matrices with a dense or a somewhere dense orbit by computing the minimal
number of generators of a dense subsemigroup (or subgroup) of the connected
component of the identity of its Zariski closure.Comment: 14 page
Representation theory for high-rate multiple-antenna code design
Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels
A new structure for difference matrices over abelian -groups
A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over -groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian -groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size in infinite families of abelian
-groups, whereas previously the largest known size was .Comment: 27 pages. Discussion of new reference [LT04
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