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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2015
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/1504.06688 |
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| _version_ | 1866912618900357120 |
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| author | Honold, Thomas Kiermaier, Michael |
| author_facet | Honold, Thomas Kiermaier, Michael |
| contents | A set $\mathcal{F}_q$ of $3$-dimensional subspaces of $\mathbb{F}_q^7$, the $7$-dimensional vector space over the finite field $\mathbb{F}_q$, is said to form a $q$-analogue of the Fano plane if every $2$-dimensional subspace of $\mathbb{F}_q^7$ is contained in precisely one member of $\mathcal{F}_q$. The existence problem for such $q$-analogues remains unsolved for every single value of $q$. Here we report on an attempt to construct such $q$-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative $q$-analogue. In particular we find a ternary subspace code of new record size $6977$, and we are able to construct a binary subspace code of the largest currently known size $329$ in an entirely computer-free manner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1504_06688 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On putative q-Analogues of the Fano Plane and Related Combinatorial Structures Honold, Thomas Kiermaier, Michael Combinatorics A set $\mathcal{F}_q$ of $3$-dimensional subspaces of $\mathbb{F}_q^7$, the $7$-dimensional vector space over the finite field $\mathbb{F}_q$, is said to form a $q$-analogue of the Fano plane if every $2$-dimensional subspace of $\mathbb{F}_q^7$ is contained in precisely one member of $\mathcal{F}_q$. The existence problem for such $q$-analogues remains unsolved for every single value of $q$. Here we report on an attempt to construct such $q$-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative $q$-analogue. In particular we find a ternary subspace code of new record size $6977$, and we are able to construct a binary subspace code of the largest currently known size $329$ in an entirely computer-free manner. |
| title | On putative q-Analogues of the Fano Plane and Related Combinatorial Structures |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1504.06688 |