Upper Bound on List-Decoding Radius of Binary Codes
Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>. For odd &l...
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| Veröffentlicht in: | IEEE transactions on information theory Jg. 62; H. 3; S. 1119 - 1128 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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IEEE
01.03.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>. For odd <inline-formula> <tex-math notation="LaTeX">L\ge 3 </tex-math></inline-formula>, an asymptotic upper bound on the rate of any such packing is proved. The resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. The method is a superposition of the linear-programming idea of Ashikhmin, Barg, and Litsyn (that was previously used to improve the estimates of Blinovsky for <inline-formula> <tex-math notation="LaTeX">L=2 </tex-math></inline-formula>) and a Ramsey-theoretic technique of Blinovsky. As an application, it is shown that for all odd <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>, the slope of the rate-radius tradeoff is zero at zero rate. |
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| AbstractList | Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most L . For odd L≥ 3 , an asymptotic upper bound on the rate of any such packing is proved. The resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. The method is a superposition of the linear-programming idea of Ashikhmin, Barg, and Litsyn (that was previously used to improve the estimates of Blinovsky for L=2 ) and a Ramsey-theoretic technique of Blinovsky. As an application, it is shown that for all odd L , the slope of the rate-radius tradeoff is zero at zero rate. Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$ . For odd $L\ge 3$ , an asymptotic upper bound on the rate of any such packing is proved. The resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. The method is a superposition of the linear-programming idea of Ashikhmin, Barg, and Litsyn (that was previously used to improve the estimates of Blinovsky for $L=2$ ) and a Ramsey-theoretic technique of Blinovsky. As an application, it is shown that for all odd $L$ , the slope of the rate-radius tradeoff is zero at zero rate. Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>. For odd <inline-formula> <tex-math notation="LaTeX">L\ge 3 </tex-math></inline-formula>, an asymptotic upper bound on the rate of any such packing is proved. The resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. The method is a superposition of the linear-programming idea of Ashikhmin, Barg, and Litsyn (that was previously used to improve the estimates of Blinovsky for <inline-formula> <tex-math notation="LaTeX">L=2 </tex-math></inline-formula>) and a Ramsey-theoretic technique of Blinovsky. As an application, it is shown that for all odd <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>, the slope of the rate-radius tradeoff is zero at zero rate. |
| Author | Polyanskiy, Yury |
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| Cites_doi | 10.1109/ISIT.2015.7282457 10.1007/s11122-005-0007-5 10.1007/978-1-4757-6048-4_20 10.1109/18.748991 10.1016/S0377-0427(98)00183-6 10.1007/11538462_27 10.1109/TIT.2005.858977 10.1006/jcta.2001.3176 10.1109/18.412711 10.1109/ISIT.2012.6283735 10.1109/TIT.1977.1055688 |
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| References | ref12 ref15 ref14 ref11 ref10 elias (ref1) 1957 polyanskiy (ref13) 2013 wozencraft (ref2) 1958 ref7 ref9 ref4 ref3 ref6 ref5 blinovsky (ref8) 1986; 22 |
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| SubjectTerms | Asymptotic methods Asymptotic properties Binary codes Combinatorial coding theory converse bounds Decoding Estimates Indexes Information theory Linear programming list-decoding Programming Slopes Thresholds Upper bound Upper bounds |
| Title | Upper Bound on List-Decoding Radius of Binary Codes |
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