Bounds on the rate of disjunctive codes

A binary code is called a superimposed cover-free (s, ℓ)-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of ℓ sets is covered by the union of s others. A binary code is called a superimposed list-decoding s L-code if the code is identified by the incidence matrix of a family of finite sets in which the union of any s sets can cover not more than L − 1 other sets of the family. For L = ℓ = 1, both of the definitions coincide and the corresponding binary code is called a superimposed s-code. Our aim is to obtain new lower and upper bounds on the rate of the given codes. The most interesting result is a lower bound on the rate of superimposed cover-free (s, ℓ)-codes based on the ensemble of constant weight binary codes. If the parameter ℓ ≥ 1 is fixed and s → ∞, then the ratio of this lower bound to the best known upper bound converges to the limit 2 e −2 = 0, 271. For the classical case ℓ = 1, the given statement means that the upper bound on the rate of superimposed s-codes obtained by A.G. Dyachkov and V.V. Rykov (1982) is asymptotically attained to within a constant factor a, 2 e −2 ≤ a ≤ 1. binary code of length N and size t. The number of 1's in column x(j), i.e., |x (j)| N i=1 x i (j), is called the weight of x(j), j ∈ [t]. 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. The standard symbol denotes the disjunct (Boolean) sum of two binary numbers: 0 0 = 0, 0 1 = 1 0 = 1 1 = 1, as well as the component-wise disjunct sum of two binary columns. We say that a column u covers column v (u v) if u v = u. The standard symbol ⌊a⌋ (⌈a⌉) will be used to denote the largest (least) integer ≤ a (≥ a).