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
Nearby in the stack
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