The satisfiability threshold for random linear equations

TL;DR

This paper determines the threshold ratio of equations to variables for the solvability of random linear systems over finite fields, providing a clearer combinatorial proof that extends understanding beyond previous methods limited to small fields.

Contribution

It introduces a new combinatorial approach to identify the satisfiability threshold for random linear equations over finite fields, generalizing beyond small fields where previous second moment methods were used.

Findings

Identified the threshold for solution existence in random linear systems over finite fields.

Provided a transparent combinatorial proof that simplifies previous complex methods.

Extended the analysis of the satisfiability threshold beyond small finite fields.

Abstract

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ 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.