2026-07-26T07:59:12Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17133 2024-03-06T10:59:03Z ddc:004 open_access

Improved Local Testing for Multiplicity Codes Karliner, Dan Ta-Shma, Amnon local testing multiplicity codes Reed Muller codes 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dan Karliner and Amnon Ta-Shma 2022 Is Part Of LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2022.11 urn:nbn:de:0030-drops-171339 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.11 eng https://creativecommons.org/licenses/by/4.0/legalcode