Construction of optimal flag codes by MRD codes
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909345052098560 |
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| author | Liu, Shuangqing Yu, Shuhui Ji, Lijun |
| author_facet | Liu, Shuangqing Yu, Shuhui Ji, Lijun |
| contents | Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-González et al. constructed optimal $(n,\mathcal{A})$-Optimum distance flag codes(ODFC) for $\mathcal {A}\subseteq \{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $k\in \mathcal A$ and $k\mid n$. In this paper, we introduce a new construction of $(n,\mathcal A)_q$-ODFCs by maximum rank-metric codes, and prove that there is an $(n,\mathcal{A})$-ODFC of size $\frac{q^n-q^{k+r}}{q^k-1}+1$ for any $\mathcal{A}\subseteq\{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $\mathcal A\cap \{k,n-k\}\neq\emptyset$, where $r\equiv n\pmod k$ and $0\leq r<k$. Furthermore, when $k>\frac{q^r-1}{q-1}$, this $(n,\mathcal A)_q$-ODFC is optimal. Specially, when $r=0$, Alonso-González et al.'s result is also obtained. We also gives a characterization of almost optimum distance flag codes, and construct a family of optimal almost optimum flag distance codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction of optimal flag codes by MRD codes Liu, Shuangqing Yu, Shuhui Ji, Lijun Discrete Mathematics 94B99 Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-González et al. constructed optimal $(n,\mathcal{A})$-Optimum distance flag codes(ODFC) for $\mathcal {A}\subseteq \{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $k\in \mathcal A$ and $k\mid n$. In this paper, we introduce a new construction of $(n,\mathcal A)_q$-ODFCs by maximum rank-metric codes, and prove that there is an $(n,\mathcal{A})$-ODFC of size $\frac{q^n-q^{k+r}}{q^k-1}+1$ for any $\mathcal{A}\subseteq\{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $\mathcal A\cap \{k,n-k\}\neq\emptyset$, where $r\equiv n\pmod k$ and $0\leq r<k$. Furthermore, when $k>\frac{q^r-1}{q-1}$, this $(n,\mathcal A)_q$-ODFC is optimal. Specially, when $r=0$, Alonso-González et al.'s result is also obtained. We also gives a characterization of almost optimum distance flag codes, and construct a family of optimal almost optimum flag distance codes. |
| title | Construction of optimal flag codes by MRD codes |
| topic | Discrete Mathematics 94B99 |
| url | https://arxiv.org/abs/2310.11285 |