Flag-Transitive Point-Primitive Quasi-Symmetric $2$-Designs and Exceptional Groups of Lie Type

Let D be a non-trivial quasi-symmetric $2$-design with two block intersection numbers $x=0$ and 2≤ y≤10, and suppose that G is an automorphism group of D. If G is flag-transitive and point-primitive, then it is known that G is either of affine type or almost simple type. In this paper, we show that the socle of G cannot be a finite simple exceptional group of Lie type.