## 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

*A**x*=

**has a solution with high probability and analyse the geometry of the set of solutions. In the special case**

*y**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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## 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