Locally testable codes with constant rate, distance, and locality

Irit Dinur; Shai Evra; Ron Livne; Alexander Lubotzky; Shahar Mozes

doi:10.1145/3519935.3520024

Locally testable codes with constant rate, distance, and locality

Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, Shahar Mozes

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester. LTCs were initially studied as important components of probabilistically checkable proofs (PCP), and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "c3-LTCs", namely LTCs with constant rate, constant distance, and constant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.

Original language	American English
Title of host publication	STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
Editors	Stefano Leonardi, Anupam Gupta
Publisher	Association for Computing Machinery
Pages	357-374
Number of pages	18
ISBN (Electronic)	9781450392648
DOIs	https://doi.org/10.1145/3519935.3520024
State	Published - 6 Sep 2022
Event	54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 - Rome, Italy Duration: 20 Jun 2022 → 24 Jun 2022

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022
Country/Territory	Italy
City	Rome
Period	20/06/22 → 24/06/22

Bibliographical note

Funding Information:
Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21. Shai Evra is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. Alexander Lubotzky’s research is supported by a grant from the Institute for Advanced Study at Princeton and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 882751). Shahar Mozes acknowledges support by ISF-Moked grant 2019/19. This material is based upon work supported by a grant from the Institute for Advanced Study.

Publisher Copyright:
© 2022 Owner/Author.

Keywords

error correcting codes
expander codes
locally testable codes

Access to Document

10.1145/3519935.3520024

Cite this

Dinur, I., Evra, S., Livne, R., Lubotzky, A., & Mozes, S. (2022). Locally testable codes with constant rate, distance, and locality. In S. Leonardi, & A. Gupta (Eds.), STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (pp. 357-374). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3519935.3520024

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abstract = "A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester. LTCs were initially studied as important components of probabilistically checkable proofs (PCP), and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist {"}c3-LTCs{"}, namely LTCs with constant rate, constant distance, and constant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.",

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note = "Funding Information: Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21. Shai Evra is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. Alexander Lubotzky{\textquoteright}s research is supported by a grant from the Institute for Advanced Study at Princeton and by the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No 882751). Shahar Mozes acknowledges support by ISF-Moked grant 2019/19. This material is based upon work supported by a grant from the Institute for Advanced Study. Publisher Copyright: {\textcopyright} 2022 Owner/Author.; 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 ; Conference date: 20-06-2022 Through 24-06-2022",

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Dinur, I, Evra, S , Livne, R, Lubotzky, A & Mozes, S 2022, Locally testable codes with constant rate, distance, and locality. in S Leonardi & A Gupta (eds), STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 357-374, 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, Rome, Italy, 20/06/22. https://doi.org/10.1145/3519935.3520024

Locally testable codes with constant rate, distance, and locality. / Dinur, Irit; Evra, Shai ; Livne, Ron et al.
STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. ed. / Stefano Leonardi; Anupam Gupta. Association for Computing Machinery, 2022. p. 357-374 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Evra, Shai

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N1 - Funding Information: Irit Dinur acknowledges support by ERC grant 772839 and ISF grant 2073/21. Shai Evra is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. Alexander Lubotzky’s research is supported by a grant from the Institute for Advanced Study at Princeton and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 882751). Shahar Mozes acknowledges support by ISF-Moked grant 2019/19. This material is based upon work supported by a grant from the Institute for Advanced Study. Publisher Copyright: © 2022 Owner/Author.

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N2 - A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester. LTCs were initially studied as important components of probabilistically checkable proofs (PCP), and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "c3-LTCs", namely LTCs with constant rate, constant distance, and constant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.

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Dinur I, Evra S , Livne R, Lubotzky A, Mozes S. Locally testable codes with constant rate, distance, and locality. In Leonardi S, Gupta A, editors, STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. 2022. p. 357-374. (Proceedings of the Annual ACM Symposium on Theory of Computing). doi: 10.1145/3519935.3520024

Locally testable codes with constant rate, distance, and locality

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Keywords

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