On Affine and Projective Failed Designs

Abstract

An affine (respectively projective) failed design $D$, denoted by $\text{AFD}(q)$ (respectively $\text{PFD}(q)$) is a configuration of $v=q^2$ points, $b=q^2+q+1$ blocks and block size $k=q$ (respectively $v=q^2+q+1$ points, $b=q^2+q+2$ blocks and block size $k=q+1$) such that every pair of points occurs in at least one block of $D$ and $D$ is minimal, that is, $D$ has no block whose deletion gives an affine plane (respectively a projective plane) of order $q$. These configurations were studied by Mendelsohn and Assaf and they conjectured that an $\text{AFD}(q)$ exists if an affine plane of order $q$ exists and a $\text{PFD}(q)$ never exists. In this paper, it is shown that an $\text{AFD}(5)$ does not exist and, therefore, the first conjecture is false in general, $\text{AFD}(q^2)$ exists if $q$ is a prime power and the second conjecture is true, that is, $\text{PFD}(q)$ never exists.