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A family of 4-designs on 26 points. (English) Zbl 0886.05038
Summary: Using the Kramer-Mesner method, $$4$$-$$(26,6,\lambda)$$ designs with $$\text{PSL} (2,25)$$ 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 $$\text{PSL} (2,25)$$, $$\text{PGL} (2,25)$$ and twisted $$\text{PGL}(2,25)$$ are being compared. It is shown that there exist $$4$$-$$(26,6,\lambda)$$ designs with $$\text{PGL}(2,25)$$, respectively twisted $$\text{PGL}(2,25)$$ 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 of automorphisms. An overview of $$t$$-$$(q+1,k,\lambda)$$ designs with $$\text{PSL}(2,q)$$ as group of automorphisms and with $$(t,k)\in \{(4,5),(4,6),(5,6)\}$$ is included.

##### MSC:
 05B30 Other designs, configurations 05B05 Combinatorial aspects of block designs
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