  A tiling of a vector space $S$ is the pair $(U,V)$ of its subsets such that
every vector in $S$ is uniquely represented as the sum of a vector from $U$ and
a vector from $V$. A tiling is connected to a perfect codes if one of the sets,
say $U$, is projective, i.e., the union of one-dimensional subspaces of $S$. A
tiling $(U,V)$ is full-rank if the affine span of each of $U$, $V$ is $S$. For
finite non-binary vector spaces of dimension at least $6$ (at least $10$), we
construct full-rank tilings $(U,V)$ with projective $U$ (both $U$ and $V$,
respectively). In particular, that construction gives a full-rank ternary
$1$-perfect code of length $13$, solving a known problem. We also discuss the
treatment of tilings with projective components as factorizations of projective
spaces.
  Keywords: perfect codes, tilings, group factorization, full-rank tilings,
projective geometry
