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Tight Bounds for Stream Decodable Error-Correcting Codes

Authors Meghal Gupta , Venkatesan Guruswami , Mihir Singhal

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Coding theory
  • Streaming computation
  • Locally decodable code
  • Lower Bounds

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Abstract

In order to communicate a message over a noisy channel, a sender (Alice) uses an error-correcting code to encode her message, a bitstring x, into a codeword. The receiver (Bob) decodes x correctly whenever there is at most a small constant fraction of adversarial errors in the transmitted codeword. We investigate the setting where Bob is restricted to be a low-space streaming algorithm. Specifically, Bob receives the message as a stream and must process it and write x in order to a write-only tape while using low (say polylogarithmic) space. Note that such a primitive then allows the execution of any downstream streaming computation on x.
We show three basic results about this setting, which are informally as follows: [(i)] 
1) There is a stream decodable code of near-quadratic length, resilient to error-fractions approaching the optimal bound of 1/4. 
2) There is no stream decodable code of sub-quadratic length, even to correct any small constant fraction of errors. 
3) If Bob need only compute a private linear function of the bits of x, instead of writing them all to the output tape, there is a stream decodable code of near-linear length.  Our constructions use locally decodable codes with additional functionality in the decoding, and (for the result on linear functions) repeated tensoring. Our lower bound, which rather surprisingly demonstrates a strong information-theoretic limitation originating from a computational restriction, proceeds via careful control of the message indices that may be output during successive blocks of the stream, a task complicated by the arbitrary state of the decoder during the algorithm.

Cite As Get BibTex

Meghal Gupta, Venkatesan Guruswami, and Mihir Singhal. Tight Bounds for Stream Decodable Error-Correcting Codes. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.13

Author Details

Meghal Gupta
  • UC Berkeley, CA, USA
Venkatesan Guruswami
  • UC Berkeley, CA, USA
Mihir Singhal
  • UC Berkeley, CA, USA

Funding

Research supported in part by NSF GRFP Fellowships (for M.G and M.S), a Simons Investigator award (for V.G), and NSF grans CCF-2210823 and CCF-2211972.

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