Conference matrices with maximum excess and two-intersection sets

A two-intersection set with parameters $(j;α,β)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $α$ or $β$ 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.

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