The satisfiability threshold for random linear equations

10/20/2017

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by   Peter Ayre, et al.

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Let A be a random m× n matrix over the finite field F_q with precisely k non-zero entries per row and let y∈ F_q^m be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system A x=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, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], 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>3. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.