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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 assignmentsconnecting 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 satisfyingassignments. Also, there exists $\epsilon_{0}\in (0,\frac{1}{k-2})$ such thatw.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-$k$ SATinstance form a single cluster.

Geometric properties of satisfying assignments of random $\epsilon$-1-in-k SAT