2,476 research outputs found
On Skew Hadamard difference sets
In this paper we construct exponentionally many non-isomorphic skew Hadamard
difference sets over an elementary abelian group of order
Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles
AbstractUsing cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets
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
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Generalized binary arrays from quasi-orthogonal cocycles
Generalized perfect binary arrays (GPBAs) were used by Jedwab to
construct perfect binary arrays. A non-trivial GPBA can exist only if its energy
is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays
(GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a
procedure to construct GOBAs based on a characterization of the arrays in terms
of 2-cocycles. As a further application, we determine negaperiodic Golay pairs
arising from generalized optimal binary sequences of small length.Junta de AndalucĂa FQM-01
Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})
Using a class of permutation polynomials of obtained from the
Ree-Tits symplectic spreads in , we construct a family of skew
Hadamard difference sets in the additive group of . With the help
of a computer, we show that these skew Hadamard difference sets are new when
and . We conjecture that they are always new when .
Furthermore, we present a variation of the classical construction of the twin
prime power difference sets, and show that inequivalent skew Hadamard
difference sets lead to inequivalent difference sets with twin prime power
parameters.Comment: 18 page
Constructions of complex Hadamard matrices via tiling Abelian groups
Applications in quantum information theory and quantum tomography have raised
current interest in complex Hadamard matrices. In this note we investigate the
connection between tiling Abelian groups and constructions of complex Hadamard
matrices. First, we recover a recent very general construction of complex
Hadamard matrices due to Dita via a natural tiling construction. Then we find
some necessary conditions for any given complex Hadamard matrix to be
equivalent to a Dita-type matrix. Finally, using another tiling construction,
due to Szabo, we arrive at new parametric families of complex Hadamard matrices
of order 8, 12 and 16, and we use our necessary conditions to prove that these
families do not arise with Dita's construction. These new families complement
the recent catalogue of complex Hadamard matrices of small order.Comment: 15 page
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