  The XOR-satisfiability (XORSAT) problem deals with a system of $n$ Boolean
variables and $m$ clauses. Each clause is a linear Boolean equation (XOR) of a
subset of the variables. A $K$-clause is a clause involving $K$ distinct
variables. In the random $K$-XORSAT problem a formula is created by choosing
$m$ $K$-clauses uniformly at random from the set of all possible clauses on $n$
variables. The set of solutions of a random formula exhibits various
geometrical transitions as the ratio $\frac{m}{n}$ varies.
  We consider a {\em coupled} $K$-XORSAT ensemble, consisting of a chain of
random XORSAT models that are spatially coupled across a finite window along
the chain direction. We observe that the threshold saturation phenomenon takes
place for this ensemble and we characterize various properties of the space of
solutions of such coupled formulae.
