Discrete & Computational Geometry

, Volume 60, Issue 2, pp 294–317

# Complex Spherical Codes with Three Inner Products

Article

## Abstract

Let X be a finite set in a complex sphere of dimension d. Let D(X) be the set of usual inner products of two distinct vectors in X. 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 wish to classify the largest possible s-codes for a given dimension d. In this paper, we consider the problem for the case $$s=3$$. In an earlier work, Roy and Suda (J Comb Des 22(3):105–148, 2014) gave certain upper bounds for the cardinality of a 3-code. 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. Further, we construct an algorithm to classify the largest 3-codes by considering representations of oriented graphs. With this algorithm, we are able to classify the largest 3-codes for dimensions 1, 2, 3 using a standard computer.

## Keywords

Complex spherical s-code s-Distance set Tight design Extremal set theory Graph representation Association scheme

05C62 05B20

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