A Steiner \(5\)-design on 36 points
Until the publication of this paper all known Steiner \(5\)-designs were on \(q+1\) points where \(q \equiv 3 \pmod{4}\), \(q\) is a prime power, and the design admits \(\text{PSL}(2,q)\) as its group of automorphisms. The authors present here a 5-\((36,6,1)\) design with automorphism group \(\text{PGL}(2,17) \times C_2\). The design is unique with this automorphism group, and even for the commutator group \(\text{PSL}(2,17) \times \text{Id}_2\) of this automorphism group there exists no further design with these parameters. The authors describe the design as a collection of \(k\)-orbits of the given group and show the incidence matrix between \(t\)-orbits and the \(k\)-orbits of the design. The design was discovered with the help of the software package DISCRETA which was developed by the authors. DISCRETA has also shown that there exist 5-\((36,6,\lambda)\) designs with this prescribed group of automorphisms for each value of \(\lambda\) in the range \(1 \leq \lambda \leq 31\). These \(t\)-designs, and many others (especially with \(t \geq 6\)) may be found at \url{http://www.mathe2.uni-bayreuth.de/betten/PUB/pub_newtdesigns_bcc.html}.
scientific article; zbMATH DE number 1392718