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Singularity of Random Integer Matrices with Large Entries

Authors Sankeerth Rao Karingula , Shachar Lovett



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Author Details

Sankeerth Rao Karingula
  • Department of Computer Science, University of California San Diego, CA, USA
Shachar Lovett
  • Department of Computer Science, University of California San Diego, CA, USA

Acknowledgements

We would like to thank Roman Vershynin and Konstantin Tikhomirov for helpful discussions.

Cite AsGet BibTex

Sankeerth Rao Karingula and Shachar Lovett. Singularity of Random Integer Matrices with Large Entries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 33:1-33:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.33

Abstract

We study the singularity probability of random integer matrices. Concretely, the probability that a random n × n matrix, with integer entries chosen uniformly from {-m,…,m}, is singular. This problem has been well studied in two regimes: large n and constant m; or large m and constant n. In this paper, we extend previous techniques to handle the regime where both n,m are large. We show that the probability that such a matrix is singular is m^{-cn} for some absolute constant c > 0. We also provide some connections of our result to coding theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Coding Theory
  • Random matrix theory
  • Singularity probability MDS codes
  • Error correction codes
  • Littlewood Offord
  • Fourier Analysis

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