  We study the geometric structure of the set of solutions of random
$\epsilon$-1-in-k SAT problem. For $l\geq 1$, two satisfying assignments $A$
and $B$ are $l$-connected if there exists a sequence of satisfying assignments
connecting them by changing at most $l$ bits at a time.
  We first prove that w.h.p. two assignments of a random $\epsilon$-1-in-$k$
SAT instance are $O(\log n)$-connected, conditional on being satisfying
assignments. Also, there exists $\epsilon_{0}\in (0,\frac{1}{k-2})$ such that
w.h.p. no two satisfying assignments at distance at least $\epsilon_{0}\cdot n$
form a "hole" in the set of assignments. We believe that this is true for all
$\epsilon >0$, and thus satisfying assignments of a random 1-in-$k$ SAT
instance form a single cluster.
