  We provide a simplified proof of the random $k$-XORSAT satisfiability
threshold theorem. As an extension we also determine the full rank threshold
for sparse random matrices over finite fields with precisely $k$ non-zero
entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao,
M\"uller: Combinatorica 2020]. The proof combines physics-inspired message
passing arguments with a surgical moment computation.
