Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery LIPIcs Resch, Nicolas Yuan, Chen Zhang, Yihan ; https://orcid.org/0000-0002-6465-6258 ddc:000 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. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2023 info:eu-repo/semantics/conferenceObject doc-type:ConferenceObject ConferenceObject http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/14083 https://research-explorer.ista.ac.at/download/14083/14091 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> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.ICALP.2023.99 info:eu-repo/semantics/altIdentifier/issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959772785 info:eu-repo/semantics/altIdentifier/arxiv/2210.07754 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess