We consider "unconstrained" random k-XORSAT, which is a uniformly random
system of m linear non-homogeneous equations in F2 over n
variables, each equation containing k≥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≥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→−∞ implies almost-sure
satisfiability, while m−n→+∞ implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in
literature survey, and improvements in presentation; removes Appendix
treating k=