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Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

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Title: Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery
Authors: Resch, Nicolas, Yuan, Chen, Zhang, Yihan
Contributors: Nicolas Resch and Chen Yuan and Yihan Zhang
Publisher Information: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Publication Year: 2023
Collection: DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics )
Subject Terms: Coding theory, List-decoding, List-recovery, Zero-rate thresholds
Description: In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of q ≥ 2. A code is called (p,L)_q-list-decodable if every radius pn Hamming ball contains less than L codewords; (p,𝓁,L)_q-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length 𝓁 and again stipulate that there be less than L codewords. Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate (p,𝓁,L)_q-list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by p_*, we in fact show that codes correcting a p_*+ε fraction of errors must have size O_ε(1), i.e., independent of n. Such a result is typically referred to as a "Plotkin bound." To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a p_*-ε fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery. Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.
Document Type: article in journal/newspaper
conference object
File Description: application/pdf
Language: English
Relation: Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.99
DOI: 10.4230/LIPIcs.ICALP.2023.99
Availability: https://doi.org/10.4230/LIPIcs.ICALP.2023.99
https://nbn-resolving.org/urn:nbn:de:0030-drops-181518
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.99
Rights: https://creativecommons.org/licenses/by/4.0/legalcode
Accession Number: edsbas.90044AA7
Database: BASE
Description
Abstract:In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of q ≥ 2. A code is called (p,L)_q-list-decodable if every radius pn Hamming ball contains less than L codewords; (p,𝓁,L)_q-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length 𝓁 and again stipulate that there be less than L codewords. Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate (p,𝓁,L)_q-list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by p_*, we in fact show that codes correcting a p_*+ε fraction of errors must have size O_ε(1), i.e., independent of n. Such a result is typically referred to as a "Plotkin bound." To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a p_*-ε fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery. Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.
DOI:10.4230/LIPIcs.ICALP.2023.99