Projective tilings and full-rank perfect codes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910483767885824 |
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| author | Krotov, Denis S. |
| author_facet | Krotov, Denis S. |
| contents | 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 |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_00105 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Projective tilings and full-rank perfect codes Krotov, Denis S. Combinatorics Discrete Mathematics 94B25, 05B45 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 |
| title | Projective tilings and full-rank perfect codes |
| topic | Combinatorics Discrete Mathematics 94B25, 05B45 |
| url | https://arxiv.org/abs/2207.00105 |