A characterization of association schemes from affine spaces (Q5926314)

An equivalence relation is called uniform if all its equivalence classes have the same size. The irreflexive part of a relation \(R\subset V\times V\) is defined as the relation \(R-I\), where \(I\) is the identity relation on \(V\). The irreflexive parts of the equivalence relations coming from all parallel classes in the finite affine space form the set of non-identity relations of an association scheme. Theorem. Let \((V,\{R_i\}_{i=0}^d)\) be a \(d\)-class association scheme, \(d>1\), such that \(R_i\cup R_0\) is an equivalence relation for all \(i\neq 0\). Then the set of all equivalence classes of these equivalence relations is the set of lines in an affine space on point set \(V\).