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Numerous computational examples suggest that if 9"(, k 1 C 9Z k arc (k 1)-and k-nets of order n, then rankp 0Z~ -rankp 9Zk_ I _> n -k + 1 ~br any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that ifP * II n, then rankt, 9Z3 > 3n -2 e, where equality holds if and only if the loop G coordinatizing 913 has a normal subloop K such that G/K is all elementary abelian group of order pC. Furthermore if n is squarefree, then mnkp 9"La = 3n -3 for every pfimep ] n, if and only if ~L3 is cyclic (i.e., ~l~z is coordinatized by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n ---2 mod 4, is in fact desarguesian of prime order.Finally, our conjectured lower bound holds with equality in the case of desarguesian ne~ (i.e., subnets of AG(2, p)), which leads to an easy description of an explicit basis for the Fl,-code of AG(2, p).

We prove the non-existence of ovoids in finite orthogonal spaces of type O 2 n ϩ 1 ( q ) for n у 4 .

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In the 1970's Paul Erdős and Dominic Welsh independently posed the problem of whether all finite partial linear spaces L are embeddable in finite projective planes. Except for the case when L has a unique embedding in a projective plane with few additional points, very little has been done which is directly applicable to this problem. In this paper it is proved that every finite partial linear space L is embeddable in a finite translation net generated by a partial spread of a vector space of even dimension. The question of whether every finite partial linear space is embedded in a finite André net is also explored. It is shown that for each positive integer n there exist finite partial linear spaces which do not embed in any André net of dimension less than or equal to n over its kernel. Mathematics Subject Classification (2000). 51E15, 51E26.