Projective tilings and full-rank perfect codes

A tiling of a finite vector space $R$ is the pair $(U,V)$ of its subsets such that $U+V=R$ and $|U|\cdot |V|=|R|$. 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 $R$. A tiling $(U,V)$ is full-rank if the affine span of each of $U$, $V$ is $R$. For 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, this construction gives a full-rank ternary $1$-perfect code of length $13$, solving a known problem. We discuss the treatment of tilings with projective components as factorizations of projective spaces.