Overlap Cycles for Steiner Quadruple Systems

Steiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is well known that Steiner quadruple systems of order v , or SQS( v ), exist if and only if v ≡ 2 , 4( mod 6 ) . Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap cycles are generalizations of universal cycles that were introduced in 2010 by Godbole, et al. Using Hanani's SQS constructions, we show that for every v ≡ 2 , 4( mod 6 ) with v > 4 there exists an SQS( v ) that admits a 1‐overlap cycle.