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New Codes on High Dimensional Expanders

Authors Irit Dinur , Siqi Liu , Rachel Yun Zhang

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • error correcting codes
  • high dimensional expanders
  • multiplication property

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Abstract

We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC). 
Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are functions on a subset of the points in 𝔽ⁿ whose restrictions to a prescribed set of affine lines has low degree. Alternatively, they are Tanner codes on high dimensional expanders, where the coordinates of the codeword correspond to triangles of a 2-dimensional expander, such that around every edge the local view forms a Reed-Solomon codeword. 
For some range of parameters our codes are provably locally testable, and their dimension is some fixed power of the block length. For another range of parameters our codes have distance and dimension that are both linear in the block length, but we do not know if they are locally testable. The codes also have the multiplication property: the coordinate-wise product of two codewords is a codeword in a related code.
The definition of the codes relies on the construction of a specific family of simplicial complexes which is a slight variant on the coset complexes of Kaufman and Oppenheim. We show a novel way to embed the triangles of these complexes into 𝔽ⁿ, with the property that links of edges embed as affine lines in 𝔽ⁿ.
We rely on this embedding to lower bound the rate of these codes in a way that avoids constraint-counting and thereby achieves non-trivial rate even when the local codes themselves have arbitrarily small rate, and in particular below 1/2.

Cite As Get BibTex

Irit Dinur, Siqi Liu, and Rachel Yun Zhang. New Codes on High Dimensional Expanders. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 27:1-27:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.27

Author Details

Irit Dinur
  • Weizmann Institute of Science, Rehovot, Israel
Siqi Liu
  • UC Berkeley, CA, USA
Rachel Yun Zhang
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Dinur, Irit: Supported by ERC grant 772839, and ISF grant 2073/21. Part of the work was done while visiting the Simons Institute for the Theory of Computing.
  • Liu, Siqi: Supported in part by the Berkeley Haas Blockchain Initiative and a donation from the Ethereum Foundation.
  • Zhang, Rachel Yun: This research was supported in part by DARPA under Agreement No. HR00112020023, an NSF grant CNS-2154149, and NSF Graduate Research Fellowship 2141064.

Acknowledgements

We wish to thank Gilles Zemor for pointing us to the work of [Hoholdt and Justesen, 2006]. We thank Amnon Ta-Shma for early comments on this work.

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