## Abstract

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is approximately locally list recoverable, as well as globally list recoverable in probabilistic near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) probabilistic near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic near-linear time global unique decoding algorithms. In the current work we obtain the following results: 1) The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic near-linear time global unique decoding algorithms. 2) If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable with query complexity and running time $N^{o(1)}$. This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity $N^{ \varepsilon }$ with rate that is exponentially small in $1/ \varepsilon $. 3) A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of $N^{\Omega (1/\log \log N)}$ on the product of query complexity and output list size for locally list recovering high-rate tensor codes.

Original language | English |
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Article number | 9195853 |

Pages (from-to) | 296-316 |

Number of pages | 21 |

Journal | IEEE Transactions on Information Theory |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2021 |

### Bibliographical note

Funding Information:Manuscript received October 7, 2019; revised July 27, 2020; accepted August 28, 2020. Date of publication September 14, 2020; date of current version December 21, 2020. The work of Swastik Kopparty was supported in part by the NSF under Grant CCF-1253886, Grant CCF-1540634, Grant CCF-1814409, and Grant CCF-1412958; and in part by the Binational Science Foundation (BSF) under Grant 2014359. The work of Nicolas Resch was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732; in part by the Natural Sciences and Engineering Research Council (NSERC) under Grant CGSD2-502898; in part by the NSF under Grant CCF-1422045, Grant CCF-1527110, Grant CCF-1618280, Grant CCF-1814603, and Grant CCF-1910588; in part by the NSF CAREER under Award CCF-1750808; in part by the Sloan Research Fellowship; and in part by the European Research Council (ERC) H2020 under Grant 74079 (ALGSTRONGCRYPTO). The work of Noga Ron-Zewi was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732, in part by the BSF under Grant 2014359, and in part by the ISF under Grant 735/20. The work of Shubhangi Saraf was supported in part by the NSF under Grant CCF-1350572, Grant CCF-1540634, and Grant CCF-1412958; in part by the BSF under Grant 2014359; in part by the Sloan Research Fellowship; and in part by the Simons Collaboration on Algorithms and Geometry. The work of Shashwat Silas was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732 and in part by the Google Fellowship in the School of Engineering at Stanford. This article was presented at the 2019 RANDOM. (Corresponding author: Shashwat Silas.) Swastik Kopparty and Shubhangi Saraf are with the Department of Mathematics, Rutgers University, New Brunswick, NJ 07102 USA, and also with the Department of Computer Science, Rutgers University, New Brunswick, NJ 07102 USA (e-mail: swastik.kopparty@gmail.com; shubhangi.saraf@gmail.com).

Funding Information:

The work of Swastik Kopparty was supported in part by the NSF under Grant CCF-1253886, Grant CCF-1540634, Grant CCF-1814409, and Grant CCF-1412958; and in part by the Binational Science Foundation (BSF) under Grant 2014359. The work of Nicolas Resch was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732; in part by the Natural Sciences and Engineering Research Council (NSERC) under Grant CGSD2-502898; in part by the NSF under Grant CCF-1422045, Grant CCF-1527110, Grant CCF-1618280, Grant CCF-1814603, and Grant CCF-1910588; in part by the NSF CAREER under Award CCF-1750808; in part by the Sloan Research Fellowship; and in part by the European Research Council (ERC) H2020 under Grant 74079 (ALGSTRONGCRYPTO). The work of Noga Ron-Zewi was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732, in part by the BSF under Grant 2014359, and in part by the ISF under Grant 735/20. The work of Shubhangi Saraf was supported in part by the NSF under Grant CCF-1350572, Grant CCF-1540634, and Grant CCF-1412958; in part by the BSF under Grant 2014359; in part by the Sloan Research Fellowship; and in part by the Simons Collaboration on Algorithms and Geometry. The work of Shashwat Silas was supported in part by the NSF-BSF under Grant CCF-1814629 and Grant 2017732 and in part by the Google Fellowship in the School of Engineering at Stanford. This article was presented at the 2019 RANDOM.

Publisher Copyright:

© 1963-2012 IEEE.

## Keywords

- Coding theory
- list-decoding and recovery
- local codes
- tensor codes

## ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences