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Explicit Time and Space Efficient Encoders Exist Only with Random Access

Authors Joshua Cook , Dana Moshkovitz

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Streaming models
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Time-Space Trade Offs
  • Error Correcting Codes
  • Encoders
  • Explicit Constructions
  • Streaming Lower Bounds
  • Sequential Access
  • Time-Space Lower Bounds
  • Lossless Condensers
  • Invertible Condensers
  • Condensers

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Abstract

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length.
In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.

Cite As Get BibTex

Joshua Cook and Dana Moshkovitz. Explicit Time and Space Efficient Encoders Exist Only with Random Access. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 5:1-5:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.5

Author Details

Joshua Cook
  • Department of Computer Science, University of Texas at Austin, TX, USA
Dana Moshkovitz
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

This material is based upon work supported by the National Science Foundation under grant number 2200956.

Acknowledgements

Thanks to Ryan Williams for questions that eventually led to this result. Thanks to David Zuckerman, Justin Oh, and Jesse Goodman for some advice about which extractors might be used in our condenser construction. Thanks to Niels Kornerup for conversations about time space lower bounds.

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