Flag-transitive symmetric designs and double quasi-symmetric designs with 8-dimensional projective orthogonal group $P{\mathrm{\Omega }}_8^{\pm }(3)$ as socle

This paper contributes to the study of flag-transitive symmetric 2- $(v,k,\lambda )$ designs that admit primitive automorphism groups whose socle is the 8-dimensional orthogonal group $P{\mathrm{\Omega }}_8^{+}(3)$ or $P{\mathrm{\Omega }}_8^{-}(3)$. We prove that no such design exists. Subsequently, we study the related double quasi-symmetric designs and provide some specific examples. The block intersection numbers of these designs are all relatively large. This paper offers a new approach to the research of double quasi-symmetric designs and symmetric partially balanced incomplete block designs.