A Comparison of \(n\)-ary Design Definitions

A generalization of (binary) balanced incomplete block designs is to allow a treatment to occur in a block more than once, that is, instead of having blocks of the design as sets, allow multisets as blocks. Such a generalization is referred to as an \(n\)-ary design. There are at least three such generalizations studied in the literature. The present note studies the relationship between these three definitions. We also give some results for the special case when \(n\) is \(3\).