Families of Finite Sets in which No Intersection of ℓ Sets Is Covered by the Union of s Others

In 1964, Kautz and Singleton (IEEE Trans. Inform. Theory10 (1964), 363–377) introduced the superimposed code concept. A binary superimposed code of strength s is identified by the incidence matrix of a family of finite sets in which no set is covered by the union of s others (J. Combin. Theory Ser. A33 (1982), 158–166 and Israel J. Math.51 (1985), 75–89). In the present paper, we consider a generalization called a binary superimposed (s,ℓ)-code which is identified by the incidence matrix of a family defined in the title. We discuss the constructions based on MDS-codes (The Theory of Error-correcting Codes, North-Holland, Amsterdam, The Netherlands, 1983) and derive upper and lower bounds on the rate of these codes.