Steiner systems 𝑆(5,6,𝑣) with 𝑣=72 and 84

Abstract

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems S ( 5 , 6 , 72 ) S(5,6,72) invariant under the group P S L 2 ( 71 ) \mathrm {PSL}_2(71) and which can be constructed using only short orbits. It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems S ( 5 , 6 , 84 ) S(5,6,84) invariant under the group P S L 2 ( 83 ) \mathrm {PSL}_2(83) and which can be constructed using only short orbits.