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Bounds for List-Decoding and List-Recovery of Random Linear Codes

Authors Venkatesan Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, Mary Wootters

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ACM Subject Classification
  • Mathematics of computing → Coding theory
Keywords
  • list-decoding
  • list-recovery
  • random linear codes
  • coding theory

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Abstract

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.

Cite As Get BibTex

Venkatesan Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters. Bounds for List-Decoding and List-Recovery of Random Linear Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.9

Author Details

Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Ray Li
  • Department of Computer Science, Stanford University, CA, USA
Jonathan Mosheiff
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Nicolas Resch
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Shashwat Silas
  • Department of Computer Science, Stanford University, CA, USA
Mary Wootters
  • Department of Computer Science, Stanford University, CA, USA

Funding

  • Guruswami, Venkatesan: Partially funded by NSF grants CCF-1563742 and CCF-1814603.
  • Li, Ray: Partially funded by NSF-CAREER grant CCF-1844628, NSF-BSF grant CCF-1814629, a Sloan Research Fellowship and NSF-GRFP grant DGE-1656518.
  • Mosheiff, Jonathan: Partially funded by NSF grants CCF-1563742 and CCF-1814603.
  • Resch, Nicolas: Partially funded by NSF grants CCF-1563742, CCF-1814603, CCF-1527110, CCF-1618280, CCF-1814603, CCF-1910588, NSF CAREER award CCF-1750808 and a Sloan Research Fellowship.
  • Silas, Shashwat: Partially funded by NSF-CAREER grant CCF-1844628, NSF-BSF grant CCF-1814629, a Sloan Research Fellowship and a Google Graduate Fellowship.
  • Wootters, Mary: Partially funded by NSF-CAREER grant CCF-1844628, NSF-BSF grant CCF-1814629 and a Sloan Research Fellowship.

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