Small quasimultiple 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) less than or equal to 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 [n+1/2] 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 Finally, we present an unpublished construction by M. Greig giving a quasidouble affine plane of order n for every positive integer n such that n - 1 and n + 1 are prime powers. (C) 1998 John Wiley & Sons, Inc.