Let A be a random m×n matrix over the finite field Fq with
precisely k non-zero entries per row and let y∈Fqm 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, 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