Association schemes of affine type over finite rings. (English) Zbl 1117.05112

Summary: W. M. Kwok [Eur. J. Comb. 13, 167–185 (1992; Zbl 0770.05100)] 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 scheme of affine type. In this paper, we define the association schemes of affine type over the finite ring \(\mathbb Z_q=\mathbb Z/q\mathbb Z\) where \(q\) is a prime power in the same manner, and calculate their character tables explicitly, using the method in A. Medrano et al. [Proc. Am. Math. Soc. 126, 701–710 (1998; Zbl 0927.05053)] and M. R. DeDeo [“Graphs over the ring of integers modulo \(2^r\)”, Ph.D. thesis, Univ. California, San Diego, CA (1998)]. 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.


05E30 Association schemes, strongly regular graphs
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