Complex spherical codes with three inner products

A finite set $X$ in a complex sphere is called an $s$-code if the number of inner products of two distinct vectors in $X$ is equal to $s$. One of major problems on $s$-codes is to classify the largest possible $3$-codes for a given dimension. A complex spherical $3$-code naturally has the structure of an oriented graph. In this paper, we study the minimal dimension which contains a complex $3$-code having the structure of a given oriented graph. It is possible for certain oriented graphs to determine the minimal dimension. The algorithm to classify the largest $3$-codes can be obtained by using this minimal dimension. The largest $3$-codes for dimensions $1$, $2$, $3$ are classified by a computer calculation based on the algorithm. Roy and Suda (2014) gave some upper bounds for the cardinality of a $3$-code, and the set attaining the bound is said to be tight. A tight code has the structure of a commutative non-symmetric association scheme. We classify tight $3$-codes for any dimension. Indeed there exists no tight $3$-code except for dimensions $1$, $2$.