Cyclic sequences of k-subsets with distinct consecutive unions

In this paper, we investigate cyclic sequences which contain as elements all k-subsets of {0, 1, . . . , n−1} exactly once such that the unions of any two consecutive k-subsets of this sequences are pairwise distinct. Furthermore, if Y is some prescribed subset of the power set of {0, 1, . . . , n− 1}, we require that all unions are in Y . In particular, we are interested in the case where Y consists of all subsets of order having the same parity as k. Among others, we show the existence of such cyclic sequences for for k = 2, 3, . . . , 7 and sufficiently large n. This kind of combinatorial problems is motivated from applications in combinatorial group testing. From our results, one obtains error detecting group testing procedures for items having the 2-consecutive positive property.