Super-simple Steiner pentagon systems

Abstract

A Steiner pentagon system of order v ( SPS ( v ) ) is said to be super-simple if its underlying ( v , 5 , 2 ) -BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary condition for the existence of a super-simple SPS ( v ) ; namely, v ⩾ 5 and v ≡ 1 or 5 ( mod 10 ) is sufficient, except for v = 5 , 15 and possibly for v = 25 . In the process, we also improve an earlier result for the spectrum of super-simple ( v , 5 , 2 ) -BIBDs, removing all the possible exceptions. We also give some new examples of Steiner pentagon packing and covering designs (SPPDs and SPCDs).