Translation planes of order 𝑞²: 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 I P ( q ) 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 I P ( q ) IP(q) enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order q 2 {q^2} which are subregular of index 3 or 4. It is conjectured (and proved for n ⩽ 3 n \leqslant 3 ) that, given a set of n disjoint circles in I P ( q ) IP(q) , the numbers of circles disjoint from each of the given n circles is asymptotic to q 3 / 2 n {q^3}/{2^n} . This conjecture, if true, would allow one to estimate the number of subregular translation planes of order q 2 {q^2} with any positive index.