Duals of Balanced Incomplete Block Designs Derived from an Affine Geometry

Abstract

It is well known at by identifying the points of an affine geometry AG (t, q) with treatments and identifying the $\mu$-flats $(1\leqq\mu<t)$ of AG (t, q) with blocks, a BIB design denoted by AG (t, q): $\mu$ is derived from AG (t, q) where q is a prime or a prime power. In this paper, we introduce a new association scheme called an affine geometrical association scheme and show that the dual of the BIB design AG (t, q): $\mu$ is an affine geometrical type PBIB design with $m=min{2\mu+1,2(t-\mu)}$ associate classes. It is also shown that in the case $\mu=1$ and $t\geqq3$, the number of the associate classes of this dual design can be reduced from three to two but it is not reducible except for the above case. From those results, we can get a new series of PBIB designs.