High Dimensional Expansion Implies Amplified Local Testability

Authors Tali Kaufman, Izhar Oppenheim

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Author Details

Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Izhar Oppenheim
  • Department of Mathematics, Ben-Gurion University of the Negev, Be'er-Sheva, Israel

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Tali Kaufman and Izhar Oppenheim. High Dimensional Expansion Implies Amplified Local Testability. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In this work, we define a notion of local testability of codes that is strictly stronger than the basic one (studied e.g., by recent works on high rate LTCs), and we term it amplified local testability. Amplified local testability is a notion close to the result of optimal testing for Reed-Muller codes achieved by Bhattacharyya et al. We present a scheme to get amplified locally testable codes from high dimensional expanders. We show that single orbit Affine invariant codes, and in particular Reed-Muller codes, can be described via our scheme, and hence are amplified locally testable. This gives the strongest currently known testability result of single orbit affine invariant codes, strengthening the celebrated result of Kaufman and Sudan.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Locally testable codes
  • High dimensional expanders
  • Amplified testing


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