Abstract:Using the Kramer-Mesner method, $4$-$(26,6,\lambda )$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda $ 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 $4$-$(26,6,\lambda )$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\{51,60,81,90,111\}$. With $\lambda $ in the set $\{60,81\}$, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda )$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \{(4,5), (4,6), (5,6)\}$ is included.
Keywords: block designs, orbits, projective linear group, projective special linear group, twisted projective linear group, Kramer-Mesner method
AMS Subject Classification: 05B30