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Bounds on the rate of disjunctive codes

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Abstract

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.

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Authors and Affiliations

  1. Probability Theory Chair, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    A. G. D’yachkov, I. V. Vorob’ev, N. A. Polyansky & V. Yu. Shchukin

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  1. A. G. D’yachkov
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  2. I. V. Vorob’ev
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  3. N. A. Polyansky
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  4. V. Yu. Shchukin
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Correspondence to A. G. D’yachkov.

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Original Russian Text © A.G. D’yachkov, I.V. Vorob’ev, N.A. Polyansky, V.Yu. Shchukin, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 1, pp. 31–63.

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D’yachkov, A.G., Vorob’ev, I.V., Polyansky, N.A. et al. Bounds on the rate of disjunctive codes. Probl Inf Transm 50, 27–56 (2014). https://doi.org/10.1134/S0032946014010037

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  • DOI: https://doi.org/10.1134/S0032946014010037

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