The Satisfiability Threshold for $k$XORSAT, using an alternative proof
Abstract
We consider "unconstrained" random $k$XORSAT, which is a uniformly random system of $m$ linear nonhomogeneous 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 satisfiability of constrained 3XORSAT, and analyzed the 2core of a random 3uniform hypergraph to extend this result to find the threshold for unconstrained 3XORSAT. 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 2core 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 $nm \to \infty$ implies almostsure satisfiability, while $mn \to \infty$ implies almostsure unsatisfiability.
 Publication:

arXiv eprints
 Pub Date:
 December 2012
 arXiv:
 arXiv:1212.3822
 Bibcode:
 2012arXiv1212.3822P
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 This version integrates the previous version's alternative proof into the paper (see arXiv:1212.1905). Other proofs are amended, and the paper's structure is clarified. The main result is improved: the phase transition occurs for an arbitrarily slowly growing gap between m and n