A simple construction of cover-free (s, ℓ)-code with certain constant weight

In the given article we generalize a construction presented in [3]. We give a method of constructing a cover-free (s, ℓ)-code. For k > s, our construction yields a (n s) ℓ × n k cover-free (s, ℓ)-code with a constant column weight. Let N , t, s and ℓ be integers, where 1 ≤ s < t and 1 < ℓ < t − s. Let denote the equality by definition, |A| – the size of the set A and [N ] {1, 2,. .. , N } – the set of integers from 1 to N. The standard symbol ⌊a⌋ will be used to denote the largest integer ≤ a. (t) (codewords) is called a binary code of length N and size t = ⌊2 RN ⌋, where a fixed parameter R > 0 is called a rate of the code X. The number of 1's in a binary column x = is called a weight of x. A code X is called a constant weight binary code of weight w, 1 ≤ w < N , if for any j ∈ [t], the weight |x (j)| = w. x i (k) = 1 for any k ∈ L. (2) 2 Main Result Let { n k } denote the family of k-subsets of set [n], and { (n s) ℓ } denote the family of non-ordered ℓ-tuples of distinct s-subsets of set [n]. For s < k < n define a binary matrix X(k, s, ℓ, n) by letting the rows and columns be, respectively, represented by the members of { (