On optimal parameters of a class of superimposed codes and designs

In 1965, Renyi suggested a combinatorial group testing model, in which the size of a testing group was restricted. In this model, Renyi considered the search of one defective element (significant factor) from the finite set of elements (factors). The corresponding optimal search designs were obtained by Katona (1966). We study Renyi's search model of several significant factors. This problem is closely related to the concept of binary superimposed codes, which were introduced by Kautz-Singleton (1964) and were investigated by D'yachkov-Rykov (1983). Our goal is to prove a lower bound on the search length and to construct the optimal superimposed codes and search designs.