Abstract
An s-subset of codewords of a binary code X is said to be \((s,\,\ell )\) -bad in X if the code X contains a subset of \(\ell \) other codewords such that the conjunction of the \(\ell \) codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called \((s,\,\ell )\) -good in X. A binary code X is said to be a cover-free (CF) \((s,\,\ell )\)-code if the code X does not contain \((s,\,\ell )\)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF \((s,\,\ell )\)-codes, namely: a binary code X is said to be an almost CF \((s,\,\ell )\)-code if the relative number of its \((s,\,\ell )\)-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 \((s,\,\ell )\)-codes is essentially greater than the rate for ordinary CF \((s,\,\ell )\)-codes.
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Acknowledgments
A. D’yachkov, I.V. Vorobyev, N.A. Polyanskii and V.Yu. Shchukin have been supported in part by the Russian Foundation for Basic Research under Grant No. 16-01-00440.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
The material in this work was presented in part at the 2015 IEEE International Symposium on Information Theory. This paper is the full paper of [8] and provides significant technical contributions over [8], e.g., proofs of all lemmas. Refer to Sect. 4.
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D’yachkov, A., Vorobyev, I., Polyanskii, N. et al. Almost cover-free codes and designs. Des. Codes Cryptogr. 82, 231–247 (2017). https://doi.org/10.1007/s10623-016-0279-3
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DOI: https://doi.org/10.1007/s10623-016-0279-3