On the existence of small quasimultiples of affine and projective planes of arbitrary order (Q805620)

Let a(n), respectively p(n) denote the minimal strictly positive integer for which there exists a \(2-(n^ 2,n,a(n))\), respectively, a \(2-n^ 2+n+1,n+1,p(n))\) design. Thus \(a(n)=p(n)=1\) if there exists a finite affine plane and (hence) a projective plane of order n. In the paper, information is presented on upper bounds for a(n) and p(n) for all \(n\leq 100\), for n of a special form (e.g., a(n), \(p(n)\leq n+2\) if n is even and n/2 is an odd prime power). The most general result is that, for n sufficiently large, a(n), \(p(n)\leq n^{29}\). It is a consequence of the (known) existence, for large n, of at least \(n^{1/14.8}\) mutually orthogonal Latin squares of order n.