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Improved Local Testing for Multiplicity Codes

Authors Dan Karliner, Amnon Ta-Shma



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Author Details

Dan Karliner
  • Department of Computer Science, Tel Aviv University, Israel
Amnon Ta-Shma
  • Department of Computer Science, Tel Aviv University, Israel

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Dan Karliner and Amnon Ta-Shma. Improved Local Testing for Multiplicity Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.11

Abstract

Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in 𝔽_p^m. Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [Karliner et al., 2022] showed that the plane test, which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees. In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM_p(m, d, s) over prime fields with arbitrary d are locally testable by an appropriate k-flat test, which tests the degree of the codeword on a random k-dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [Karliner et al., 2022] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [Haramaty et al., 2013]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Error-correcting codes
Keywords
  • local testing
  • multiplicity codes
  • Reed Muller codes

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References

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