154 research outputs found
Coherent configurations and triply regular association schemes obtained from spherical designs
Delsarte-Goethals-Seidel showed that if is a spherical -design with
degree satisfying , carries the structure of an association
scheme. Also Bannai-Bannai showed that the same conclusion holds if is an
antipodal spherical -design with degree satisfying . As a
generalization of these results, we prove that a union of spherical designs
with a certain property carries the structure of a coherent configuration. We
derive triple regularity of tight spherical -designs, mutually unbiased
bases, linked symmetric designs with certain parameters.Comment: 17page
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
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
Complex spherical codes with three inner products
Let be a finite set in a complex sphere of dimension. Let be
the set of usual inner products of two distinct vectors in . A set is
called a complex spherical -code if the cardinality of is and
contains an imaginary number. We would like to classify the largest
possible -codes for given dimension . In this paper, we consider the
problem for the case . Roy and Suda (2014) gave a certain upper bound for
the cardinalities of -codes. A -code is said to be tight if
attains the bound. We show that there exists no tight -code except for
dimensions , . Moreover we make an algorithm to classify the largest
-codes by considering representations of oriented graphs. By this algorithm,
the largest -codes are classified for dimensions , , with a
current computer.Comment: 26 pages, no figur
Symmetric and skew-symmetric -matrices with large determinants
We show that the existence of -matrices having largest possible
determinant is equivalent to the existence of certain tournament matrices. In
particular, we prove a recent conjecture of Armario. We also show that large
submatrices of conference matrices are determined by their spectrum.Comment: 14 page
- …