429 research outputs found
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
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
Symmetric Bush-type generalized Hadamard matrices and association schemes
We define Bush-type generalized Hadamard matrices over abelian groups and
construct symmetric Bush-type generalized Hadamard matrices over the additive
group of finite field , a prime power. We then show and study
an association scheme obtained from such generalized Hadamard matrices.Comment: 10 page
- β¦