— We introduce and discuss the concept of a binary superimposed (s,)-code identified by a family of finite sets in which no intersection of sets is covered by the union of s others. Upper and lower bounds on the rate of these codes are formulated. Their proofs will be given in [7]. Several constructions of these codes are considered in the second part of the present paper [6]. Integers N and t are called the length and size of code C, respectively. 1 0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0, 1 ∧ 1 = 1. We say that vector x is covered by vector y if x y = y. Remark. Obviously, definition 1 is equivalent to the condition: j∈L x(j) is not covered by j ∈S x(j). An interpretation of an (s,)-code as a family of set with certain properties is given in " Part II " of the present paper [6]. We call an element p ∈ P(s, , t) a positive supersets, and an element P ∈ p — a positive set in terms of superset p. 2 Background and Motivations For the special case = 1, a superimposed (s, 1)-code ((s, 1)-design) is called a superimposed s-code (s-design). They were introduced in [1] and studied in [2, 3, 4]. See also the book [5]. Superimposed (s,)-codes and designs arise from the problem of group testing for supersets, which can be stated as follows. Assume that we have a set of t objects (we identify them by integers j ∈ [t]), in which several subsets P 1 ,. .. , P k ⊂ [t] are positive. Assume that a number of positive subsets k ≤ s, and the size of each positive subset is 2 not greater then. Our aim is to determine all positive subsets using a finite number of tests. In each test we take a group G ⊂ [t] and examine it. The test result r(G) = 1 (positive) if P m ⊆ G for some m ∈ [k], and r(G) = 0 (negative) otherwise.) be N testing groups. In the current model we use nonadaptive testing, which means that we select all groups before any test is performed. Let vector r = r(G) (r(G 1),. .. , r(G N)) represent the results of N tests. Encode the testing groups by the set C = {x(1),. .. , …