Cover-free Codes Research Articles

Almost cover-free codes and designs

An s-subset of codewords of a binary code X is said to be $$(s,\,\ell )$$(s,l)-bad in X if the code X contains a subset of $$\ell $$l other codewords such that the conjunction of the $$\ell $$l codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called $$(s,\,\ell )$$(s,l)-good in X. A binary code X is said to be a cover-free (CF) $$(s,\,\ell )$$(s,l)-code if the code X does not contain $$(s,\,\ell )$$(s,l)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF $$(s,\,\ell )$$(s,l)-codes, namely: a binary code X is said to be an almost CF $$(s,\,\ell )$$(s,l)-code if the relative number of its $$(s,\,\ell )$$(s,l)-good s-subsets is close to 1. We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes. Our main result shows that the capacity for almost CF $$(s,\,\ell )$$(s,l)-codes is essentially greater than the rate for ordinary CF $$(s,\,\ell )$$(s,l)-codes.

Cover-free codes and separating system codes

We give some relations between the asymptotic rates of cover-free (CF) codes, separating system (SS) codes and completely separating system (CSS) codes. We also provide new upper bounds on the asymptotic rate of SS codes based on known results for CF and CSS codes. Finally, we derive a random coding bound for the asymptotic rate of SS codes and give tables of numerical values corresponding to our improved upper bounds.

Bounds on the rate of disjunctive codes

A binary code is said to be a disjunctive (s, l) cover-free code if it is an incidence matrix of a family of sets where the intersection of any l sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L ? 1 other sets of this family. For L = l = 1, both definitions coincide, and the corresponding binary code is called a disjunctive s-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive (s, l) cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as s ? ?, with an arbitrary fixed l ? 1, to the limit 2e?2 = 0.271 ... In the classical case of l = 1, this means that the upper bound on the rate of disjunctive s-codes constructed in 1982 by D'yachkov and Rykov is asymptotically attained up to a constant factor a, 2e?2 ≤ a ≤ 1.

Separating codes and a new combinatorial search model

We propose a new group testing model, which is related to separating codes and cover-free codes.

On the construction of (w, r) cover-free codes

We construct a new class of concatenated cover-free codes. We prove that in this class there exists a sequence of (w, r) cover-free codes which has a nonzero limit rate for w, r = const. We consider application of cover-free codes to key distribution systems.

A Note on the Uniqueness of (w, r) Cover-Free Codes

A binary code is called a (w, r) cover-free code if it is the incidence matrix of a family of sets where the intersection of any w sets is not covered by the union of any other r sets. For certain (w, r) cover-free codes with a simple structure, we obtain a new condition of optimality and uniqueness up to row and/or column permutations.

On the Optimality of Trivial (w, r)-Cover-Free Codes

A (w, r)-cover-free code is the incidence matrix of a family of sets where no intersection of w members of the family is covered by the union of r others. We obtain a new condition in view of which (w, r)-cover-free codes with a simple structure are optimal. We also introduce (w, r)-cover-free codes with a constraint set.

Asymptotic Upper Bound for the Rate of (w, r) Cover-Free Codes

A binary code is called a (w, r) cover-free code if it is the incidence matrix of a family of sets where the intersection of any w of the sets is not covered by the union of any other r sets. Such a family is called a (w, r) cover-free family. We obtain a new recurrent inequality for the rate of (w, r) cover-free codes, which improves previously known upper bounds on the rate.

Application of cover-free codes and combinatorial designs to two-stage testing

We study combinatorial and probabilistic properties of cover-free codes and block designs which are useful for their efficient application as the first stage of two-stage group testing procedures. Particular attention is paid to these procedures because of their importance in such applications as monoclonal antibody generation and cDNA library screening.

Application of cover-free codes and combinatorial designs to two-stage testing

Abstract We study the theory and design of efficient combinatorial and probabilistic group testing procedures. Particular attention is paid to two-stage tests because of their importance in such applications as monoclonal antibody generation and cDNA library screening.