Complex spherical codes with three inner products

TL;DR

This paper investigates the maximum size of complex spherical 3-codes with three inner products, proving non-existence of tight codes beyond certain dimensions and classifying largest codes in low dimensions using an algorithm.

Contribution

It proves the non-existence of tight 3-codes in dimensions greater than 2 and develops an algorithm to classify largest 3-codes in low dimensions.

Findings

No tight 3-codes exist for dimensions > 2.

Largest 3-codes classified for dimensions 1, 2, 3.

Algorithm based on oriented graph representations used for classification.

Abstract

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.