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 π½_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.
@InProceedings{karliner_et_al:LIPIcs.APPROX/RANDOM.2022.11, author = {Karliner, Dan and Ta-Shma, Amnon}, title = {{Improved Local Testing for Multiplicity Codes}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {11:1--11:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.11}, URN = {urn:nbn:de:0030-drops-171339}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.11}, annote = {Keywords: local testing, multiplicity codes, Reed Muller codes} }
Feedback for Dagstuhl Publishing