A block-transitive t-(k^2,k,lambda) design with socle PSL(2,q), q>=4, can exist only for q=8, and then it is a 2-(36,6,lambda) design with lambda in {2,6,9,12,18,36}.
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Block-transitive t-(k^2,k,\lambda) designs associated to two dimensional projective special linear groups
A block-transitive t-(k^2,k,lambda) design with socle PSL(2,q), q>=4, can exist only for q=8, and then it is a 2-(36,6,lambda) design with lambda in {2,6,9,12,18,36}.