Upper Bound on List-Decoding Radius of Binary Codes

• Published in: IEEE transactions on information theory Vol. 62; no. 3; pp. 1119 - 1128
• Main Author: Polyanskiy, Yury
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 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; Combinatorial coding theory; list-decoding; converse bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2016.2516560

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.