{"author":[{"last_name":"Resch","first_name":"Nicolas","full_name":"Resch, Nicolas"},{"full_name":"Yuan, Chen","last_name":"Yuan","first_name":"Chen"},{"id":"2ce5da42-b2ea-11eb-bba5-9f264e9d002c","full_name":"Zhang, Yihan","orcid":"0000-0002-6465-6258","last_name":"Zhang","first_name":"Yihan"}],"language":[{"iso":"eng"}],"day":"01","year":"2023","abstract":[{"text":"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.\r\nOur 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.\r\nTechnically, 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.","lang":"eng"}],"oa":1,"scopus_import":"1","volume":261,"date_published":"2023-07-01T00:00:00Z","quality_controlled":"1","article_number":"99","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1","publication_identifier":{"isbn":["9783959772785"],"issn":["1868-8969"]},"ddc":["000"],"intvolume":"       261","title":"Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery","citation":{"mla":"Resch, Nicolas, et al. “Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery.” <i>50th International Colloquium on Automata, Languages, and Programming</i>, vol. 261, 99, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023, doi:<a href=\"https://doi.org/10.4230/LIPIcs.ICALP.2023.99\">10.4230/LIPIcs.ICALP.2023.99</a>.","ista":"Resch N, Yuan C, Zhang Y. 2023. Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery. 50th International Colloquium on Automata, Languages, and Programming. ICALP: International Colloquium on Automata, Languages, and Programming, LIPIcs, vol. 261, 99.","ieee":"N. Resch, C. Yuan, and Y. Zhang, “Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery,” in <i>50th International Colloquium on Automata, Languages, and Programming</i>, Paderborn, Germany, 2023, vol. 261.","apa":"Resch, N., Yuan, C., &#38; Zhang, Y. (2023). Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery. In <i>50th International Colloquium on Automata, Languages, and Programming</i> (Vol. 261). Paderborn, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.ICALP.2023.99\">https://doi.org/10.4230/LIPIcs.ICALP.2023.99</a>","chicago":"Resch, Nicolas, Chen Yuan, and Yihan Zhang. “Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery.” In <i>50th International Colloquium on Automata, Languages, and Programming</i>, Vol. 261. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. <a href=\"https://doi.org/10.4230/LIPIcs.ICALP.2023.99\">https://doi.org/10.4230/LIPIcs.ICALP.2023.99</a>.","ama":"Resch N, Yuan C, Zhang Y. Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery. In: <i>50th International Colloquium on Automata, Languages, and Programming</i>. Vol 261. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2023. doi:<a href=\"https://doi.org/10.4230/LIPIcs.ICALP.2023.99\">10.4230/LIPIcs.ICALP.2023.99</a>","short":"N. Resch, C. Yuan, Y. Zhang, in:, 50th International Colloquium on Automata, Languages, and Programming, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023."},"article_processing_charge":"Yes","publication":"50th International Colloquium on Automata, Languages, and Programming","external_id":{"arxiv":["2210.07754"]},"license":"https://creativecommons.org/licenses/by/4.0/","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","alternative_title":["LIPIcs"],"publication_status":"published","date_created":"2023-08-20T22:01:13Z","month":"07","file":[{"access_level":"open_access","creator":"dernst","success":1,"date_updated":"2023-08-21T07:23:18Z","file_size":1141497,"date_created":"2023-08-21T07:23:18Z","checksum":"a449143fec3fbebb092cb8ef3b53c226","file_name":"2023_LIPIcsICALP_Resch.pdf","content_type":"application/pdf","file_id":"14091","relation":"main_file"}],"_id":"14083","conference":{"name":"ICALP: International Colloquium on Automata, Languages, and Programming","start_date":"2023-07-10","end_date":"2023-07-14","location":"Paderborn, Germany"},"status":"public","type":"conference","date_updated":"2023-08-21T07:26:01Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"MaMo"}],"oa_version":"Published Version","doi":"10.4230/LIPIcs.ICALP.2023.99","acknowledgement":"Nicolas Resch: Research supported in part by ERC H2020 grant No.74079 (ALGSTRONGCRYPTO). Chen Yuan: Research supported in part by the National Key Research and Development Projects under Grant 2022YFA1004900 and Grant 2021YFE0109900, the National Natural Science Foundation of China under Grant 12101403 and Grant 12031011.\r\nAcknowledgements YZ is grateful to Shashank Vatedka, Diyuan Wu and Fengxing Zhu for inspiring discussions.","file_date_updated":"2023-08-21T07:23:18Z"}