We consider regular Hjelmslev planes satisfying additional assumptions; in particular, we always assume the planes to be balanced. The main result of the paper is the determination of the spectra of invariants of regular n-uniform H-planes and of balanced minimally uniform regular H-planes. These are determined exactly (not only up to orders of projective planes), the only other spectrum known exactly up to now being the spectrum of desarguesian H-planes. We also prove that our construction method for balanced minimally uniform regular H-planes is canonical.
O. INTRODUCTION
One of the outstanding problems in the theory of finite Hjelmslev planes is the determination of all possible invariants of such planes. This is in general much too hard; e.g. it includes the determination of all orders of projective planes. Thus one considers the spectra of certain special classes of Hjelmslev planes. Only three results have been obtained in that direction, all of which are quite deep:
(i) The spectrum of desarguesian Hjelmslev planes is the set of all (q% q), q a prime power (Klingenberg [11], Bacon [3]);
(ii) The spectrum of n-uniform Hjelmslev planes is the set of all (r "-1, r), r the order of a projective plane (Artmann [2], using T6rner [13]);
(iii) The spectrum of all balanced]-minimally uniform Hjelmslev planes is the set of Lenz pairs (Drake-T6rner [7] and Drake ).
Note that (ii) and (iii) are only determined up to orders of projective planes.
In this paper, we will determine two more spectra completely (not only up to projective planes).
After reviewing some material on Hjelmslev and Klingenberg planes in Part 1, we consider balanced H-matrices in Part 2 and n-uniform regular Hjelmslev planes in Part 3. The main result there will be:
(iv) The spectrum of regular n-uniform Hjelmslev planes is the set of all (q,-1, q), q a prime power.
We then study balanced minimally uniform regular Hjelmslev planes in Part 4 and obtain * The author acknowledges the hospitality of the University of Florida while doing this research. t Called 'regular' in . As this interferes with the use of regular in [8], we here choose the term 'balanced' instead. We choose this term, because balanced H-planes are PBIBD's (oral communication by D. Drake).