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

翻译：向量空间$S$的铺砌是指其子集对$(U,V)$，使得$S$中的每个向量均可唯一表示为$U$中向量与$V$中向量之和。若其中一个集合（例如$U$）是射影的，即$S$的一维子空间之并，则该铺砌与完美码相关联。铺砌$(U,V)$称为满秩的，若$U$和$V$各自的仿射张成空间均为$S$。对于维度至少为$6$（或至少为$10$）的有限非二元向量空间，我们构造了具有射影$U$（或$U$与$V$均为射影）的满秩铺砌$(U,V)$。特别地，该构造给出了长度为$13$的满秩三元$1$-完美码，解决了一个已知问题。我们还将具有射影分量的铺砌视为射影空间的因式分解进行讨论。关键词：完美码，铺砌，群分解，满秩铺砌，射影几何