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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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