  A two-intersection set with parameters $(j;\alpha,\beta)$ for a block design
is a $j$-subset of the point set of the design, which intersects every block in
$\alpha$ or $\beta$ points. In this paper, we show the existence of a
two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block
design obtained from translations of the set of nonzero squares in the finite
field of order $q=4m^2+1$. As an application, we give a construction of
conference matrices with maximum excess based on the two-intersection sets.
