论文标题
投影砖和完整的完美代码
Projective tilings and full-rank perfect codes
论文作者
论文摘要
矢量太空$ s $的平铺是其子集的$ $（u，v）$，因此$ s $中的每个向量都以$ u $的矢量和v $ v $的矢量为代表。如果其中一种（例如$ u $）为投影，即$ s $的一维子空间的结合，则瓷砖连接到完美的代码。如果$ u $，$ v $的仿射跨度为$ s $，则瓷砖$（u，v）$是全等级。对于尺寸有限的非二进制矢量空间，至少$ 6 $（至少$ 10 $），我们用投影$ u $（分别$ u $和$ u $和$ v $）构建了全级瓷砖$（u，v）$。特别是，该构建提供了全级三元$ 1 $ - 完美的长度代码$ 13 $，解决了已知问题。我们还讨论了用投影成分作为投影空间的因素化的瓷砖处理。 关键字：完美的代码，砖砌，群体分解，全等级砖，投影几何形状
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
