A family of 4-designs on 38 points

Using the Kramer-Mesner method, 4-(26, 6, λ) designs with PSL(2, 25) as a group of automorphisms and with λ in the set {30, 51, 60, 81, 90, 111} are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups PSL(2, 25), PGL(2, 25) and twisted PGL(2, 25) are being compared. It is shown that there exist 4-(26, 6, λ) designs with PGL(2, 25), respectively twisted PGL(2, 25) as a group of automorphisms and with λ in the set {51, 60, 81, 90, 111}. With λ in the set {60, 81}, there exist designs which possess all three considered groups as groups of automorphisms. An overview of t-(q + 1, k, λ) designs with PSL(2, q) as group of automorphisms and with (t, k) ∈ {(4, 5), (4, 6), (5, 6)} is included.