More on the existence of small quasimultiples of affine and projective planes of arbitrary order (Q2725042)

There has been some interest in constructing \(s\)-fold quasimultiples of projective and affine planes, that is designs \(S_s(2,n+1;n^2+n+1)\) resp.\ \(S_s(2,n;n^2)\), of arbitrary order \(n\) (even in the cases where no projective or affine plane of that order exists). The smallest possible values for \(s\) are denoted by \(p(n)\) and \(a(n)\), respectively. As shown by the reviewer [On the existence of small quasimultiples of affine and projective planes of arbitrary order. II., J. Comb. Des. 3, No. 6, 427-432 (1995; Zbl 0885.05024)], one has \(a(n),p(n) < n^{10}\) for all sufficiently large \(n\). The author substantially improves this result (using a different approach) by establishing \(a(n),p(n) < n^3\) for all \(n\).