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Polynomials, Divided Differences, and Codes

Author S. Venkitesh

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Author Details

S. Venkitesh
  • Tel Aviv University, Israel

Funding

A part of this work was done while the author was supported in part by BSF grant 2021683, ISF grant 735/20, and by the European Union (ERC, ECCC, 101076663).
  • Venkitesh, S.: Len Blavatnik and the Blavatnik Family Foundation.

Acknowledgements

The author thanks Dean Doron, Mrinal Kumar, Noga Ron-Zewi, Amnon Ta-Shma, and Mary Wootters for patiently listening to and giving feedback on presentations of this work at different times. A part of this work was done while the author was a postdoc at the Department of Computer Science, University of Haifa, and was participating in the program on Error Correcting Codes and Computation (2024) at the Simons Institute for the Theory of Computing, UC Berkeley.

Cite As Get BibTex

S. Venkitesh. Polynomials, Divided Differences, and Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 93:1-93:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.93

Abstract

Multiplicity codes (Kopparty et al., J. ACM 2014) are multivariate polynomial codes where the codewords are described by evaluations of polynomials (with a degree bound) and their derivatives up to some order (the multiplicity parameter), on a suitably chosen affine set of points. While efficient decoding algorithms were known in some special cases of point sets, by a reduction to univariate multiplicity codes, a general algorithm for list decoding up to the distance of the code when the point set is an arbitrary finite grid, was obtained only recently (Bhandari et al., IEEE TIT 2023). This required the characteristic of the field to be zero or larger than the degree bound, which is somewhat necessary since list decoding up to distance with small output list size is not possible when the characteristic is significantly smaller than the degree.
In this work, we present an alternative construction based on divided differences of polynomials, that conceptually resembles the classical multiplicity codes but has good list decodability "insensitive to the field characteristic". We obtain a simple algorithm that list decodes this code up to distance for arbitrary finite grids over all finite fields. Our construction can also be interpreted as a folded Reed-Muller code, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Theory of computation → Error-correcting codes
Keywords
  • Error-correcting code
  • polynomial code
  • Reed-Solomon code
  • Reed-Muller code
  • folded Reed-Solomon code
  • folded Reed-Muller code
  • multiplicity code
  • divided difference
  • q-derivative
  • polynomial method
  • list decoding
  • list decoding capacity
  • linear algebraic list decoding

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