×

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.

MSC:

05E30 Association schemes, strongly regular graphs
PDF BibTeX XML Cite
Full Text: DOI EuDML Link

References:

[1] R. Baeza, Quadratic forms over semilocal rings. Springer 1978. · Zbl 0382.10014
[2] E. Bannai, Character tables of commutative association schemes. In: Finite geometries, buildings, and related topics (Pingree Park, CO, 1988), 105-128, Oxford Univ. Press 1990.
[3] E. Bannai, T. Ito, Algebraic combinatorics.I. Benjamin/Cummings Publishing Co. 1984.
[4] Bannai E., Mem. Fac. Sci. Kyushu Univ. Ser. 44 pp 129– (1990)
[5] B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi sums.Wiley-Interscience 1998. · Zbl 0906.11001
[6] A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs.Springer 1989. · Zbl 0747.05073
[7] M. R. DeDeo, Graphs over the ring of integers modulo 2r. PhD thesis, U.C.S.D., CA, 1998.
[8] Knebusch M., Math. Z. 108 pp 255– (1969)
[9] W., European J. Combin. 13 pp 167– (1992)
[10] H. Matsumura, Commutative ring theory. Cambridge Univ. Press 1986. · Zbl 0603.13001
[11] Medrano A., J. Comput. Appl. Math. 68 pp 221– (1996)
[12] Medrano A., Proc. Amer. Math. Soc. 126 pp 701– (1998)
[13] A. Munemasa, The geometry of orthogonal groups over finite fields. JSPS-DOST Lecture Notes in Math. 3, Sophia University, Tokyo, 1996.
[14] Salie H., Z. 34 pp 91– (1931)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.