Translation planes of order $q^{2}$: asymptotic estimates

Abstract:

R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane $IP(q)$. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in $IP(q)$ enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order ${q^2}$ which are subregular of index 3 or 4. It is conjectured (and proved for $n \leqslant 3$) that, given a set of n disjoint circles in $IP(q)$, the numbers of circles disjoint from each of the given n circles is asymptotic to ${q^3}/{2^n}$. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order ${q^2}$ with any positive index.