@inproceedings{11fd2c2004cb4f27be693a94a72ffc3f,
title = "Improved Local Testing for Multiplicity Codes",
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 Fmp. Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [6] 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 MRMp(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 [6] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [5]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.",
keywords = "Reed Muller codes, local testing, multiplicity codes",
author = "Dan Karliner and Amnon Ta-Shma",
note = "Publisher Copyright: {\textcopyright} Dan Karliner and Amnon Ta-Shma.; 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 ; Conference date: 19-09-2022 Through 21-09-2022",
year = "2022",
month = sep,
day = "1",
doi = "https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.11",
language = "الإنجليزيّة",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
editor = "Amit Chakrabarti and Chaitanya Swamy",
booktitle = "Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022",
address = "ألمانيا",
}