Flag-transitive point-primitive quasi-symmetric 2-designs with block intersection numbers 0 and y ≤ 10

Abstract

In this paper, we show that for a non-trivial quasi-symmetric 2-design $D$ with two block intersection numbers $x=0$ and $2\le y\le 10$, if $G\le \mathrm{Aut}\left(D\right)$ is flag-transitive and point-primitive, then G is either of affine type or almost simple type. Moreover, we prove that the socle of G cannot be an alternating group. If the socle of G is a sporadic group, then $D$ and G must be one of the following: $D$ is a 2-$(12,6,5)$ design with block intersection numbers $0,3$ and $G=M_{11}$, or $D$ is a 2-$(22,6,5)$ design with block intersection numbers $0,2$ and $G=M_{22}$ or $M_{22}:2$.