Coherent configurations and triply regular association schemes obtained from spherical designs

Delsarte-Goethals-Seidel showed that if $X$ is a spherical $t$-design with degree $s$ satisfying $t\geq 2s-2$, $X$ carries the structure of an association scheme. Also Bannai-Bannai showed that the same conclusion holds if $X$ is an antipodal spherical $t$-design with degree $s$ satisfying $t=2s-3$. 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 $4,5,7$-designs, mutually unbiased bases, linked symmetric designs with certain parameters.Comment: 17page

Complex spherical codes with two inner products

A finite set $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly regular tournaments, or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric $D$-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 $\mathbb{F}_q$, $q$ 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 $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$ contains an imaginary number. We would like to classify the largest possible $s$-codes for given dimension $d$. In this paper, we consider the problem for the case $s=3$. Roy and Suda (2014) gave a certain upper bound for the cardinalities of $3$-codes. A $3$-code $X$ is said to be tight if $X$ attains the bound. We show that there exists no tight $3$-code except for dimensions $1$, $2$. Moreover we make an algorithm to classify the largest $3$-codes by considering representations of oriented graphs. By this algorithm, the largest $3$-codes are classified for dimensions $1$, $2$, $3$ with a current computer.Comment: 26 pages, no figur

Symmetric and skew-symmetric {0,±1}\{0,\pm 1\}-matrices with large determinants

We show that the existence of $\{\pm 1\}$-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