Nested \(BIBDs\) from Affine Planes

Abstract

For any prime power \(q\), there exists an affine plane of order \(q\). The complement of an affine plane is a balanced incomplete block design (BIBD) with block size \(q^2-q\). In this note, a proof is given that the blocks can be split into sub-blocks to form a nested BIBD with parameters \((q^2, q^2+q, q^3+q^2, q^2-1,q-1)\). Alternatively, this is a generalized tournament design with one game each round, involving \(q\) teams, each team with \(q-1\) players.