Divisible minimal codes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918045665984512 |
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| author | Chubenko, Vladimir Kurz, Sascha |
| author_facet | Chubenko, Vladimir Kurz, Sascha |
| contents | Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$ is denoted by $m(k,q)$. Here we determine $m(7,2)$, $m(8,2)$, and $m(9,2)$, as well as full classifications of all codes attaining $m(k,2)$ for $k\le 7$ and those attaining $m(9,2)$. We give improved upper bounds for $m(k,2)$ for all $10\le k\le 17$. It turns out that in many cases the attaining extremal codes have the property that the weights of all codewords are divisible by some constant $Δ>1$. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by $Δ$. As a byproduct we also give a few binary linear codes improving the best known lower bound for the minimum distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00885 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Divisible minimal codes Chubenko, Vladimir Kurz, Sascha Combinatorics Information Theory 94B05 (51E23) Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$ is denoted by $m(k,q)$. Here we determine $m(7,2)$, $m(8,2)$, and $m(9,2)$, as well as full classifications of all codes attaining $m(k,2)$ for $k\le 7$ and those attaining $m(9,2)$. We give improved upper bounds for $m(k,2)$ for all $10\le k\le 17$. It turns out that in many cases the attaining extremal codes have the property that the weights of all codewords are divisible by some constant $Δ>1$. So, here we study the minimum lengths of minimal codes where we additionally assume that the weights of the codewords are divisible by $Δ$. As a byproduct we also give a few binary linear codes improving the best known lower bound for the minimum distance. |
| title | Divisible minimal codes |
| topic | Combinatorics Information Theory 94B05 (51E23) |
| url | https://arxiv.org/abs/2312.00885 |