The Satisfiability Threshold For Random Linear Equations

Abstract

Let A be a random m × n matrix over the finite field \(\mathbb{F}_q\) with precisely k non-zero entries per row and let \(y\in\mathbb{F}_q^m\) be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system Ax = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3], and the proof technique was subsequently extended to the cases q = 3,4 [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to q > 4. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

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Acknowledgment

We thank Charilaos Efthymiou, Alan Frieze, Mike Molloy and Wesley Pegden for helpful discussions.

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Correspondence to Pu Gao.

Additional information

Gao’s research is supported by ARC DE170100716 and ARC DP160100835.

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Ayre, P., Coja-Oghlan, A., Gao, P. et al. The Satisfiability Threshold For Random Linear Equations. Combinatorica 40, 179–235 (2020). https://doi.org/10.1007/s00493-019-3897-3

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Mathematics Subject Classification (2010)

  • 05C80
  • 05C50