Improved Local Testing for Multiplicity Codes

Dan Karliner, Amnon Ta-Shma

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022
EditorsAmit Chakrabarti, Chaitanya Swamy
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772495
DOIs
StatePublished - 1 Sep 2022
Event25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 - Virtual, Urbana-Champaign, United States
Duration: 19 Sep 202221 Sep 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume245

Conference

Conference25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022
Country/TerritoryUnited States
CityVirtual, Urbana-Champaign
Period19/09/2221/09/22

Keywords

  • Reed Muller codes
  • local testing
  • multiplicity codes

All Science Journal Classification (ASJC) codes

  • Software

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