Sho Suda

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et al. proposed an association scheme of 64 points in R^1^4 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

Coherent configurations and triply regular association schemes obtained from spherical designs

Delsarte, Goethals and Seidel showed that if X is a spherical t-design with degree s satisfying t>=2s-2, X carries the structure of an association scheme. Also Bannai and 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 systems of symmetric designs with certain parameters.

A generalization of the Erdős-Ko-Rado theorem to t-designs in certain semilattices

The Erdos-Ko-Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.

Symmetric and Skew‐Symmetric {0,±1}‐Matrices with Large Determinants

We show that the existence of

Complex spherical codes with two inner products

\{\pm 1\}

Mutually Unbiased Biangular Vectors and Association Schemes

-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.

Linked systems of symmetric group divisible designs

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.

A strongly regular decomposition of the complete graph and its association scheme

A class of unbiased (−1, 1)-matrices extracted from a single Hadamard matrix is shown to provide uniform imprimitive association schemes of four class and six class.

New parameters of subsets in polynomial association schemes

We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.

A semidefinite programming approach to a cross-intersection problem with measures

Abstract For any positive integer m, the complete graph on 2 2 m ( 2 m + 2 ) vertices is decomposed into 2 m + 1 commuting strongly regular graphs, which give rise to a symmetric association scheme of class 2 m + 2 − 2 . Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.