Small quasimultiple of affine and projective planes; some improved bounds

We improve the known bounds on r(n):=min{lambda | an (n^2,n,lambda)-RBIBD exists} in the case where n+1 is a prime power. In such a case r(n) is proved to be at most n+1. If, in addition, n+1 is the product of twin prime powers, then r(n)<=n/2. We also improve the known bounds on p(n):=min{lambda | an (n^2+n+1,n+1,lambda)-BIBD exists} in the case where n^2+n+1 is a prime power. In such a case p(n) is bounded at worst by $\lfloor{n+1\over2}\rfloor$ but better bounds could be obtained exploiting the multiplicative structure of GF(n^2+n+1). Finally, we present an unpublished construction by M. Greig giving a quasidouble affine plane of order n for every positive integer n such that n-1 and n+1 are prime powers.