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).
1. Introduction
A k-net of order n is an incidence structure consisting of n 2 points and nk distinguished subsets called lines, such that