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On generalized Moore geometries. I, II. (English) Zbl 0626.51006
The author finds a bound for the diameter d of a nontrivial generalized Moore geometry.
Using formulas from E. Bannai and T. Ito [“Algebraic Combinatorics. I: Association schemes.” (1984; Zbl 0555.05019)], in the first part he proves that the minimal polynomial must factor into factors of degree at most 3 over integers. Then he applies the rationalization method of R. M. Damerell and M. A. Georgiacodis [J. Lond. Math. Soc., II. Ser. 23, 1-9 (1981; Zbl 0467.05019)] followed by reduction \(modulo 2\) and \(modulo 3.\) He obtains a list of exponential diophantine equations, each of which turns out to have a finite number of solutions. The largest of these solutions is \(d=161.\)
In the second part the diameter d of a generalized Moore geometry is shown to be at most 13. He applies various ad hoc methods on a case - by - case method to lower the bound. The author means that the bound 13 could be further reduced, although it seems difficult.
Reviewer: P.Burda

MSC:
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
51E99 Finite geometry and special incidence structures
05B25 Combinatorial aspects of finite geometries
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