文章基本信息

标题：Bounds for List-Decoding and List-Recovery of Random Linear Codes
本地全文：下载
作者：Venkatesan Guruswami ; Ray Li ; Jonathan Mosheiff 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：176
页码：9:1-9:21
DOI：10.4230/LIPIcs.APPROX/RANDOM.2020.9
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：A family of error-correcting codes is list-decodable from error fraction p if, for every code in the family, the number of codewords in any Hamming ball of fractional radius p is less than some integer L that is independent of the code length. It is said to be list-recoverable for input list size ð" if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than ð" values. The parameter L is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size L â' â^Z, is known to be 1-h_q(p) for list-decoding, and 1-log_q ð" for list-recovery, where q is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below q is the alphabet size, and Îµ > 0 is the gap to capacity). - A random linear code of rate 1 - log_q(ð") - Îµ requires list size L â¥ ð"^{Î©(1/Îµ)} for list-recovery from input list size ð". This is surprisingly in contrast to completely random codes, where L = O(ð"/Îµ) suffices w.h.p. - A random linear code of rate 1 - h_q(p) - Îµ requires list size L â¥ âOS {h_q(p)/Îµ+0.99}âO< for list-decoding from error fraction p, when Îµ is sufficiently small. - A random binary linear code of rate 1 - hâ,,(p) - Îµ is list-decodable from average error fraction p with list size with L â¤ âOS {hâ,,(p)/Îµ}âO< + 2. (The average error version measures the average Hamming distance of the codewords from the center of the Hamming ball, instead of the maximum distance as in list-decoding.) The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values. Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset - or some symbol-wise permutation of it - lies in a random linear code with high probability. This uses a recent characterization of (Mosheiff, Resch, Ron-Zewi, Silas, Wootters, 2019) of configurations of codewords that are contained in random linear codes. Our upper bound follows from a refinement of the techniques of (Guruswami, HÃ¥stad, Sudan, Zuckerman, 2002) and strengthens a previous result of (Li, Wootters, 2018), which applied to list-decoding rather than average-radius list-decoding.
关键词：list-decoding; list-recovery; random linear codes; coding theory