Divisibility of Griesmer Codes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913885852794880 |
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| author | Deng, Haihua Huang, Hexiang Xiang, Qing |
| author_facet | Deng, Haihua Huang, Hexiang Xiang, Qing |
| contents | In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if $p^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. In this paper, we show that if $q^e|d$, then $C$ has a basis consisting of $k$ codewords such that the first $\min\left\{e+1,k\right\}$ of them span a Griesmer subcode with constant weight $d$ and any $k-1$ of them span a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. Using the $p$-adic algebraic method together with this basis, we prove that if $q^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. Based on this fact, using the geometric approach with the aforementioned basis, we show that if $p^e|d$, then $Δ|{\rm wt}(c)$ for all $c\in C$, where $Δ=\left\lceil p^{e-(f-1)(q-2)}\right\rceil$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07846 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divisibility of Griesmer Codes Deng, Haihua Huang, Hexiang Xiang, Qing Combinatorics 11T71, 11A07, 94B05 In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if $p^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. In this paper, we show that if $q^e|d$, then $C$ has a basis consisting of $k$ codewords such that the first $\min\left\{e+1,k\right\}$ of them span a Griesmer subcode with constant weight $d$ and any $k-1$ of them span a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. Using the $p$-adic algebraic method together with this basis, we prove that if $q^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. Based on this fact, using the geometric approach with the aforementioned basis, we show that if $p^e|d$, then $Δ|{\rm wt}(c)$ for all $c\in C$, where $Δ=\left\lceil p^{e-(f-1)(q-2)}\right\rceil$. |
| title | Divisibility of Griesmer Codes |
| topic | Combinatorics 11T71, 11A07, 94B05 |
| url | https://arxiv.org/abs/2506.07846 |