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Pseudorandom Linear Codes Are List-Decodable to Capacity

Authors Aaron (Louie) Putterman , Edward Pyne

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Author Details

Aaron (Louie) Putterman
  • Harvard University, Cambrdige, MA, USA
Edward Pyne
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Putterman, Aaron (Louie): Supported in party by the Simons Investigator Fellowship of Boaz Barak, NSF grant DMS-2134157, DARPA grant W911NF2010021, and DOE grant DE-SC0022199. Supported in part by the Simons Investigator Award of Salil Vadhan and Simons Investigator Award of Madhu Sudan and NSF Award CCF 2152413.
  • Pyne, Edward: Supported by an Akamai Presidential Fellowship.

Acknowledgements

We would like to thank Madhu Sudan for helpful conversations.

Cite AsGet BibTex

Aaron (Louie) Putterman and Edward Pyne. Pseudorandom Linear Codes Are List-Decodable to Capacity. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 90:1-90:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.90

Abstract

We introduce a novel family of expander-based error correcting codes. These codes can be sampled with randomness linear in the block-length, and achieve list decoding capacity (among other local properties). Our expander-based codes can be made starting from any family of sufficiently low-bias codes, and as a consequence, we give the first construction of a family of algebraic codes that can be sampled with linear randomness and achieve list-decoding capacity. We achieve this by introducing the notion of a pseudorandom puncturing of a code, where we select n indices of a base code C ⊂ 𝔽_q^m in a correlated fashion. Concretely, whereas a random linear code (i.e. a truly random puncturing of the Hadamard code) requires O(n log(m)) random bits to sample, we sample a pseudorandom linear code with O(n + log (m)) random bits by instantiating our pseudorandom puncturing as a length n random walk on an exapnder graph on [m]. In particular, we extend a result of Guruswami and Mosheiff (FOCS 2022) and show that a pseudorandom puncturing of a small-bias code satisfies the same local properties as a random linear code with high probability. As a further application of our techniques, we also show that pseudorandom puncturings of Reed-Solomon codes are list-recoverable beyond the Johnson bound, extending a result of Lund and Potukuchi (RANDOM 2020). We do this by instead analyzing properties of codes with large distance, and show that pseudorandom puncturings still work well in this regime.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • error-correcting codes

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References

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