Primitive commutative association schemes with a non-symmetric relation of valency 3 (Q1976318)

Let \(Y=(X,\{R_i\}_{0\leq i\leq d})\) be an association scheme. If \(X=F_q\) (a finite field of order \(q\)), \(K\) is a subgroup of \(F_q^\times\) and \(R_i=\{(x,y)\mid y-x\in a_iK\}\) (\(1\leq i\leq d\)) where \(\{a_i\}_{1\leq i\leq d}\) is the transversal of \(F_q^\times\) by \(K\), then this scheme is called cyclotomic and denoted \(\text{Cyc}(q,|K|)\). Theorem 1.5. Let \(Y=(X,\{R_i\}_{0\leq i\leq d})\) be a primitive commutative association scheme. If there is a non-symmetric relation \(R_i\) with valency 3, then the cardinality of \(X\) is equal to either \(p\) or \(p^2\) where \(p\) is an odd prime. Moreover, if \(|X|=p\) then \(Y\simeq \text{Cyc}(p,3)\), and if \(|X|=p^2\) then there exists a relation isomorphic to a non-diagonal relation of \(\text{Cyc}(p^2,3)\).