  We consider "unconstrained" random $k$-XORSAT, which is a uniformly random
system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$
variables, each equation containing $k \geq 3$ 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 satisfiability
of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform
hypergraph to extend this result to find the threshold for unconstrained
3-XORSAT.
  We show that $m/n=1$ remains a sharp threshold for satisfiability of
constrained $k$-XORSAT for every $k\ge 3$, and we use standard results on the
2-core of a random $k$-uniform hypergraph to extend this result to find the
threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow
the phase transition window, showing that $m-n \to -\infty$ implies almost-sure
satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.
