Erasures versus errors in local decoding and property testing

Sofya Raskhodnikova; Noga Ron-Zewi; Nithin Varma

doi:10.1002/rsa.21031

Erasures versus errors in local decoding and property testing

Sofya Raskhodnikova, Noga Ron-Zewi, Nithin Varma

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. We first investigate local list-decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list-decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list-decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich–Levin theorem. We further study approximate locally erasure list-decodable codes and use them to construct a property that is erasure-resiliently testable with query complexity independent of the input length, (Formula presented.), but requires (Formula presented.) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.

Original language	English
Pages (from-to)	640-670
Number of pages	31
Journal	Random Structures and Algorithms
Volume	59
Issue number	4
DOIs	https://doi.org/10.1002/rsa.21031
State	Published - Dec 2021

Bibliographical note

Funding Information:
The authors express their gratitude to anonymous reviewers whose comments helped improve the presentation of this article. The authors are thankful to Venkatesan Guruswami for helping to tighten the analysis of the local erasure list‐decoder for the Hadamard code and also for making a suggestion that led to Observation 5.2 . The authors are grateful to Prahladh Harsha, Or Meir, Ramesh Krishnan S. Pallavoor, Adam Smith, Sergey Yekhanin, and Avi Wigderson for useful discussions. Last but not least, the authors would like to thank the sponsors and organizers of the Workshop on Local Algorithms 2018 for making this collaboration possible. The first author was supported by National Science Foundation (NSF) grants CCF‐142297, CCF‐1832228, and CCF‐1909612. The second author was supported in part by the Israel Science Foundation (ISF) grant 735/20. Most of this work was done when the third author was a student at the Boston University, where he was supported by NSF grants CCF‐142297, and CCF‐1832228. The third author was also supported by the ISF grant 497/17, and by the PBC Fellowship for Postdoctoral Fellows by the Israeli Council of Higher Education.

Publisher Copyright:
© 2021 Wiley Periodicals LLC.

Keywords

Goldreich–Levin theorem
Hadamard code
erasures versus errors
local decoding
property testing

ASJC Scopus subject areas

Software
General Mathematics
Computer Graphics and Computer-Aided Design
Applied Mathematics

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10.1002/rsa.21031

Cite this

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title = "Erasures versus errors in local decoding and property testing",

abstract = "We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. We first investigate local list-decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list-decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list-decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich–Levin theorem. We further study approximate locally erasure list-decodable codes and use them to construct a property that is erasure-resiliently testable with query complexity independent of the input length, (Formula presented.), but requires (Formula presented.) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.",

keywords = "Goldreich–Levin theorem, Hadamard code, erasures versus errors, local decoding, property testing",

author = "Sofya Raskhodnikova and Noga Ron-Zewi and Nithin Varma",

note = "Funding Information: The authors express their gratitude to anonymous reviewers whose comments helped improve the presentation of this article. The authors are thankful to Venkatesan Guruswami for helping to tighten the analysis of the local erasure list‐decoder for the Hadamard code and also for making a suggestion that led to Observation 5.2 . The authors are grateful to Prahladh Harsha, Or Meir, Ramesh Krishnan S. Pallavoor, Adam Smith, Sergey Yekhanin, and Avi Wigderson for useful discussions. Last but not least, the authors would like to thank the sponsors and organizers of the Workshop on Local Algorithms 2018 for making this collaboration possible. The first author was supported by National Science Foundation (NSF) grants CCF‐142297, CCF‐1832228, and CCF‐1909612. The second author was supported in part by the Israel Science Foundation (ISF) grant 735/20. Most of this work was done when the third author was a student at the Boston University, where he was supported by NSF grants CCF‐142297, and CCF‐1832228. The third author was also supported by the ISF grant 497/17, and by the PBC Fellowship for Postdoctoral Fellows by the Israeli Council of Higher Education. Publisher Copyright: {\textcopyright} 2021 Wiley Periodicals LLC.",

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N2 - We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure-resilient and tolerant property testing. We first investigate local list-decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list-decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list-decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich–Levin theorem. We further study approximate locally erasure list-decodable codes and use them to construct a property that is erasure-resiliently testable with query complexity independent of the input length, (Formula presented.), but requires (Formula presented.) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.

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Erasures versus errors in local decoding and property testing

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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