Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery
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| Název: | Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery |
|---|---|
| Autoři: | Resch, Nicolas, Yuan, Chen, Zhang, Yihan |
| Přispěvatelé: | Nicolas Resch and Chen Yuan and Yihan Zhang |
| Informace o vydavateli: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik |
| Rok vydání: | 2023 |
| Sbírka: | DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics ) |
| Témata: | Coding theory, List-decoding, List-recovery, Zero-rate thresholds |
| Popis: | 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. |
| Druh dokumentu: | article in journal/newspaper conference object |
| Popis souboru: | application/pdf |
| Jazyk: | 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 |
| Dostupnost: | 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 |
| Přístupové číslo: | edsbas.90044AA7 |
| Databáze: | BASE |
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