On generalized Moore geometries. I, II (Q580707)

The author finds a bound for the diameter d of a nontrivial generalized Moore geometry. Using formulas from \textit{E. Bannai} and \textit{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 \textit{R. M. Damerell} and \textit{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.