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Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries

Authors Gil Cohen , Tal Yankovitz

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Relaxed locally decodable codes
  • Relxaed locally correctable codes
  • RLCC
  • RLDC

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Abstract

Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log^{69} n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^{e+o(1)} exist.
In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)^{2+o(1)}. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by "vanilla" expander codes which indeed have the necessary property: local testability in the code’s vicinity.

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Gil Cohen and Tal Yankovitz. Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.8

Author Details

Gil Cohen
  • Department of Computer Science, Tel Aviv University, Israel
Tal Yankovitz
  • Department of Computer Science, Tel Aviv University, Israel

Funding

The research leading to these results has received funding from the ERC starting grant 949499 and from the Israel Science Foundation grant 1569/18.

Acknowledgements

We are grateful to Marcel Dall'Agnol and Pedro Paredes for identifying an inaccuracy in our original proof, which has been corrected in this revision.

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