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A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been considered in the past, we provide a new, simple Gray code for fixed-weight m-ary strings. In addition, we consider a relatively new type of Gray code known as overlap cycles and prove basic existence results concerning overlap cycles for fixed-weight and weight-range m-ary words. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is s-overlap cycles, which relax such a constraint. In this paper we study weak orders, which are relations that are transitive and complete. We prove the existence of universal and s-overlap cycles for weak orders, as well as for fixed height and/or weight weak orders, and apply the results to cycles for ordered partitions. [ABSTRACT FROM AUTHOR]

Abstract: Open problems arising from the Workshop on Generalizations of de Bruijn cycles and Gray Codes at the Banff International Research Station in December, 2004. [Copyright &y& Elsevier]