---
_id: '14083'
abstract:
- lang: eng
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,\U0001D4C1,L)_q-list-recoverability
is a generalization where we place radius pn Hamming balls on every point of a
combinatorial rectangle with side length \U0001D4C1 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,\U0001D4C1,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."
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."
alternative_title:
- LIPIcs
article_number: '99'
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Nicolas
full_name: Resch, Nicolas
last_name: Resch
- first_name: Chen
full_name: Yuan, Chen
last_name: Yuan
- first_name: Yihan
full_name: Zhang, Yihan
id: 2ce5da42-b2ea-11eb-bba5-9f264e9d002c
last_name: Zhang
orcid: 0000-0002-6465-6258
citation:
ama: 'Resch N, Yuan C, Zhang Y. Zero-rate thresholds and new capacity bounds for
list-decoding and list-recovery. In: 50th International Colloquium on Automata,
Languages, and Programming. Vol 261. Schloss Dagstuhl - Leibniz-Zentrum für
Informatik; 2023. doi:10.4230/LIPIcs.ICALP.2023.99'
apa: 'Resch, N., Yuan, C., & Zhang, Y. (2023). Zero-rate thresholds and new
capacity bounds for list-decoding and list-recovery. In 50th International
Colloquium on Automata, Languages, and Programming (Vol. 261). Paderborn,
Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ICALP.2023.99'
chicago: Resch, Nicolas, Chen Yuan, and Yihan Zhang. “Zero-Rate Thresholds and New
Capacity Bounds for List-Decoding and List-Recovery.” In 50th International
Colloquium on Automata, Languages, and Programming, Vol. 261. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2023. https://doi.org/10.4230/LIPIcs.ICALP.2023.99.
ieee: N. Resch, C. Yuan, and Y. Zhang, “Zero-rate thresholds and new capacity bounds
for list-decoding and list-recovery,” in 50th International Colloquium on Automata,
Languages, and Programming, Paderborn, Germany, 2023, vol. 261.
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: Automata, Languages and Programming, LIPIcs,
vol. 261, 99.'
mla: Resch, Nicolas, et al. “Zero-Rate Thresholds and New Capacity Bounds for List-Decoding
and List-Recovery.” 50th International Colloquium on Automata, Languages, and
Programming, vol. 261, 99, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2023, doi:10.4230/LIPIcs.ICALP.2023.99.
short: N. Resch, C. Yuan, Y. Zhang, in:, 50th International Colloquium on Automata,
Languages, and Programming, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2023.
conference:
end_date: 2023-07-14
location: Paderborn, Germany
name: 'ICALP: Automata, Languages and Programming'
start_date: 2023-07-10
corr_author: '1'
date_created: 2023-08-20T22:01:13Z
date_published: 2023-07-01T00:00:00Z
date_updated: 2025-09-08T08:31:53Z
day: '01'
ddc:
- '000'
department:
- _id: MaMo
doi: 10.4230/LIPIcs.ICALP.2023.99
external_id:
arxiv:
- '2210.07754'
file:
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checksum: a449143fec3fbebb092cb8ef3b53c226
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creator: dernst
date_created: 2023-08-21T07:23:18Z
date_updated: 2023-08-21T07:23:18Z
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intvolume: ' 261'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
publication: 50th International Colloquium on Automata, Languages, and Programming
publication_identifier:
isbn:
- '9783959772785'
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
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scopus_import: '1'
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title: Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
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