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List Decoding Quotient Reed-Muller Codes

Authors Omri Gotlib , Tali Kaufman, Shachar Lovett

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Reed-Muller Codes
  • Quotient Code
  • Quotient Reed-Muller Code
  • List Decoding
  • High Rank Variety
  • High-Order Fourier Analysis
  • Error-Correcting Codes

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Abstract

Reed-Muller codes consist of evaluations of n-variate polynomials over a finite field 𝔽 with degree at most d. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if and only if it satisfies all degree-d constraints.
For a subset X̃ ⊆ 𝔽ⁿ, we introduce the notion of X̃-quotient Reed-Muller code. A function F:X̃ → 𝔽 is a valid codeword in the quotient code if it satisfies all the constraints of degree-d polynomials lying in X̃. This gives rise to a novel phenomenon: a quotient codeword may have many extensions to original codewords. This weakens the connection between original codewords and quotient codewords which introduces a richer range of behaviors along with substantial new challenges.
Our goal is to answer the following question: what properties of X̃ will imply that the quotient code inherits its distance and list-decoding radius from the original code? We address this question using techniques developed by Bhowmick and Lovett [Abhishek Bhowmick and Shachar Lovett, 2014], identifying key properties of 𝔽ⁿ used in their proof and extending them to general subsets X̃ ⊆ 𝔽ⁿ. By introducing a new tool, we overcome the novel challenge in analyzing the quotient code that arises from the weak connection between original and quotient codewords. This enables us to apply known results from additive combinatorics and algebraic geometry [David Kazhdan and Tamar Ziegler, 2018; David Kazhdan and Tamar Ziegler, 2019; Amichai Lampert and Tamar Ziegler, 2021] to show that when X̃ is a high rank variety, X̃-quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.

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Omri Gotlib, Tali Kaufman, and Shachar Lovett. List Decoding Quotient Reed-Muller Codes. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 1:1-1:44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.1

Author Details

Omri Gotlib
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Shachar Lovett
  • Department of Computer Science and Engineering, UC San Diego, CA, USA

Funding

  • Kaufman, Tali: Supported by ISF.
  • Lovett, Shachar: supported by NSF award 2425349 and a Simons investigator award.

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