The Satisfiability Threshold fork-XORSAT

Abstract

We consider ‘unconstrained’ randomk-XORSAT, which is a uniformly random system ofmlinear non-homogeneous equations in$\mathbb{F}$2overnvariables, each equation containingk⩾ 3 variables, and also consider a ‘constrained’ model where every variable appears in at least two equations. Dubois and Mandler proved thatm/n= 1 is a sharp threshold for satisfiability of constrained 3-XORSAT, and analysed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT.We show thatm/n= 1 remains a sharp threshold for satisfiability of constrainedk-XORSAT for everyk⩾ 3, and we use standard results on the 2-core of a randomk-uniform hypergraph to extend this result to find the threshold for unconstrainedk-XORSAT. For constrainedk-XORSAT we narrow the phase transition window, showing thatm − n→ −∞ implies almost-sure satisfiability, whilem − n→ +∞ implies almost-sure unsatisfiability.