Association schemes of affine type over finite rings

Kwok [10] studied the association schemes obtained by the action of the semidirect products of the orthogonal groups over the finite fields and the underlying vector spaces. They are called the assiciation schemes of affine type. In this paper, we define the association schemes of affine type over the finite ring ℤ_q = ℤ/qℤ where q is a prime power in the same manner, and calculate their character tables explicitly, using the method in Medrano et al. [13] and DeDeo [8]. In particular, it turns out that the character tables are described in terms of the Kloosterman sums. We also show that these association schemes are self-dual.