The Full Rank Condition for Sparse Random Matrices

Authors Amin Coja-Oghlan, Jane Gao, Max Hahn-Klimroth, Joon Lee, Noela Müller, Maurice Rolvien

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Amin Coja-Oghlan
  • Department of Computer Science, TU Dortmund, Germany
Jane Gao
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Max Hahn-Klimroth
  • Department of Computer Science, TU Dortmund, Germany
Joon Lee
  • Communication Theory Laboratory, École Polytechnique Fédérale de Lausanne, Switzerland
Noela Müller
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
Maurice Rolvien
  • Department of Computer Science, TU Dortmund, Germany

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Amin Coja-Oghlan, Jane Gao, Max Hahn-Klimroth, Joon Lee, Noela Müller, and Maurice Rolvien. The Full Rank Condition for Sparse Random Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments.

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Theory of computation → Error-correcting codes
  • random matrices
  • rank
  • finite fields
  • rationals


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