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We consider "unconstrained" random $k$-XORSAT, which is a uniformly randomsystem of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$variables, each equation containing $k \ge 3$ variables, and also consider a"constrained" model where every variable appears in at least two equations.Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiabilityof constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniformhypergraph to extend this result to find the threshold for unconstrained3-XORSAT.  We show that $m/n=1$ remains a sharp threshold for satisfiability ofconstrained $k$-XORSAT for every $k \ge 3$, and we use standard results on the2-core of a random $k$-uniform hypergraph to extend this result to find thethreshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrowthe phase transition window, showing that $n-m \to \infty$ implies almost-suresatisfiability, while $m-n \to \infty$ implies almost-sure unsatisfiability.

The Satisfiability Threshold for $k$-XORSAT, using an alternative proof