Complex Spherical Codes with Three Inner Products
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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.
KeywordsComplex spherical s-code s-Distance set Tight design Extremal set theory Graph representation Association scheme
Mathematics Subject Classification05C62 05B20
Hiroshi Nozaki is supported by JSPS KAKENHI Grant Numbers 25800011, 26400003, 16K17569, 17K0515501. Sho Suda is supported by JSPS KAKENHI Grant Numbers 15K21075, 26400003, 17K0515501. The authors thank an anonymous referee for some useful comments and suggestions.
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