## Abstract

We strengthen the notion of double samplers, first introduced by Dinur and Kaufman [''High dimensional expanders imply agreement expanders,"" in Proc. 58th IEEE Symp. on Foundations of Comp. Science, IEEE, 2017, pp. 974-985], which are samplers with additional combinatorial properties, and whose existence we prove using high-dimensional expanders. The ABNNR code construction [N. Alon et al., IEEE Trans. Inform. Theory, 38 (1992), pp. 509-516] achieves large distance by starting with a base code C with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler, then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code C but rather to any string. In this case the resulting code is approximate-list-decodable, i.e., the output list contains an approximation to the original input. Our list-decoding algorithm works as follows: It uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list-decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high-dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

Original language | English |
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Pages (from-to) | 301-349 |

Number of pages | 49 |

Journal | SIAM Journal on Computing |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:\ast Received by the editors July 24, 2019; accepted for publication (in revised form) November 11, 2020; published electronically March 9, 2021. A preliminary version of this paper appeared in Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2019 [8]. https://doi.org/10.1137/19M1276650 Funding: The work of the first and fourth authors was supported by ERC-CoG grant 772839. The work of the second author was supported by the Department of Atomic Energy, Government of India, under project 12-R\&D-TFR-5.01-0500 and in part by UGC-ISF grant and the Swarnajayanti Fellowship. The work of the third author was supported by a BSF grant and an ERC grant. The work of the fifth author was supported by ISF grant 952/18. \dagger Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel (irit.dinur@ weizmann.ac.il, inbal.livni@weizmann.ac.il). \ddagger School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India (prahladh@tifr.res.in). \S Bar-Ilan University, Ramat Gan, Israel (kaufmant@mit.edu). \P Tel Aviv University, Tel Aviv, Israel (amnon@tau.ac.il).

Publisher Copyright:

© 2021 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license

## Keywords

- Error correcting codes
- Expander graphs
- High-dimensional expanders
- List decoding
- Pseudorandomness