Reduction for block-transitive t-\((k^2,k,\lambda )\) designs

In this paper, we study block-transitive automorphism groups of t- \((k^2,k,\lambda )\) $(k^2,k,\lambda )$ designs. We prove that a block-transitive automorphism group G of a t- \((k^2,k,\lambda )\) $(k^2,k,\lambda )$ design must be point-primitive, and G is either an affine group or an almost simple group. Moreover, the nontrivial t- \((k^2,k,\lambda )\) $(k^2,k,\lambda )$ designs admitting block-transitive automorphism groups of almost simple type with sporadic socle and alternating socle are classified.