Incidence matrices of finite attenuated spaces and class dimension of association schemes

Let d , k , n be integers with 1 ≤ d < k ≤ n − d . In Kantor (1972), Kantor proved that the incidence matrix of d -dimensional subspaces versus k -dimensional subspaces of an n -dimensional vector space has full row rank over the real number field R . In this paper, we generalize Kantor’s result to the attenuated space A ( n + ℓ ; F q ) and show that the incidence matrix of d -dimensional subspaces versus k -dimensional subspaces of A ( n + ℓ ; F q ) also has full row rank over R . As an application, we obtain upper bounds for the class dimension of association schemes based on attenuated spaces.