Complex spherical codes with two inner products

TL;DR

This paper characterizes tight complex spherical 2-codes using combinatorial structures like doubly regular tournaments and skew Hadamard matrices, and explores their maximal configurations related to skew-symmetric D-optimal designs.

Contribution

It provides a characterization of tight complex spherical 2-codes via combinatorial objects and determines their embedding dimensions using eigenvalue multiplicities.

Findings

Characterization of tight complex spherical 2-codes through doubly regular tournaments and skew Hadamard matrices.

Identification of maximal 2-codes related to skew-symmetric D-optimal designs.

Determination of the smallest embedding dimension via eigenvalue multiplicities of the Seidel matrix.

Abstract

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.