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Bounds for List-Decoding and List-Recovery of Random Linear Codes Guruswami, Venkatesan Li, Ray Mosheiff, Jonathan Resch, Nicolas Silas, Shashwat Wootters, Mary list-decoding list-recovery random linear codes coding theory 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 𝓁 if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than 𝓁 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 → ∞, is known to be 1-h_q(p) for list-decoding, and 1-log_q 𝓁 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 ε > 0 is the gap to capacity). - A random linear code of rate 1 - log_q(𝓁) - ε requires list size L ≥ 𝓁^{Ω(1/ε)} for list-recovery from input list size 𝓁. This is surprisingly in contrast to completely random codes, where L = O(𝓁/ε) suffices w.h.p. - A random linear code of rate 1 - h_q(p) - ε requires list size L ≥ ⌊ {h_q(p)/ε+0.99}⌋ for list-decoding from error fraction p, when ε is sufficiently small. - A random binary linear code of rate 1 - h₂(p) - ε is list-decodable from average error fraction p with list size with L ≤ ⌊ {h₂(p)/ε}⌋ + 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, Håstad, Sudan, Zuckerman, 2002) and strengthens a previous result of (Li, Wootters, 2018), which applied to list-decoding rather than average-radius list-decoding. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Venkatesan Guruswami and Ray Li and Jonathan Mosheiff and Nicolas Resch and Shashwat Silas and Mary Wootters 2020 Is Part Of LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2020.9 urn:nbn:de:0030-drops-126126 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.9 eng https://creativecommons.org/licenses/by/3.0/legalcode