On the Generalized Covering Radii of Reed-Muller Codes

Dor Elimelech, Hengjia Wei, Moshe Schwartz

Research output: Contribution to journalArticlepeer-review

Abstract

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters.

Original languageEnglish
JournalIEEE Transactions on Information Theory
DOIs
StateAccepted/In press - 1 Jan 2022

Keywords

  • Codes
  • Generators
  • Hamming weight
  • Linear codes
  • Protocols
  • Reed-Muller codes
  • Reed-Muller codes
  • Upper bound
  • covering algorithm
  • generalized covering radius

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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