Hermitian Rank Metric Codes and Duality
In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-linear matrix codes in the ambient space <inline-formula&g...
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| Format: | Article |
| Language: | English |
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IEEE
2021-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/9371673/ |
| _version_ | 1811209795146874880 |
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| author | Javier De La Cruz Jorge Robinson Evilla Ferruh Ozbudak |
| author_facet | Javier De La Cruz Jorge Robinson Evilla Ferruh Ozbudak |
| author_sort | Javier De La Cruz |
| collection | DOAJ |
| description | In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-linear matrix codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> and for both <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula>-linear codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>. Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-duality between bases of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula> and prove that a <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-self dual basis exists if and only if <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is an odd integer. We obtain connections on the dual codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula>. Furthermore, we present connections between Hermitian <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and Euclidean <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>. |
| first_indexed | 2024-04-12T04:44:56Z |
| format | Article |
| id | doaj.art-a654d99fbeb04460b98a11fb10056147 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-12T04:44:56Z |
| publishDate | 2021-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-a654d99fbeb04460b98a11fb100561472022-12-22T03:47:31ZengIEEEIEEE Access2169-35362021-01-019384793848710.1109/ACCESS.2021.30645039371673Hermitian Rank Metric Codes and DualityJavier De La Cruz0Jorge Robinson Evilla1Ferruh Ozbudak2https://orcid.org/0000-0002-1694-9283Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, ColombiaDepartamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, ColombiaDepartment of Mathematics, Institute of Applied Mathematics, Middle East Technical University, Ankara, TurkeyIn this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-linear matrix codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> and for both <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula>-linear codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>. Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-duality between bases of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula> and prove that a <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-self dual basis exists if and only if <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is an odd integer. We obtain connections on the dual codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula>. Furthermore, we present connections between Hermitian <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and Euclidean <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>.https://ieeexplore.ieee.org/document/9371673/Rank metric codesadditive rank metric codesHermitian rank metric codes |
| spellingShingle | Javier De La Cruz Jorge Robinson Evilla Ferruh Ozbudak Hermitian Rank Metric Codes and Duality IEEE Access Rank metric codes additive rank metric codes Hermitian rank metric codes |
| title | Hermitian Rank Metric Codes and Duality |
| title_full | Hermitian Rank Metric Codes and Duality |
| title_fullStr | Hermitian Rank Metric Codes and Duality |
| title_full_unstemmed | Hermitian Rank Metric Codes and Duality |
| title_short | Hermitian Rank Metric Codes and Duality |
| title_sort | hermitian rank metric codes and duality |
| topic | Rank metric codes additive rank metric codes Hermitian rank metric codes |
| url | https://ieeexplore.ieee.org/document/9371673/ |
| work_keys_str_mv | AT javierdelacruz hermitianrankmetriccodesandduality AT jorgerobinsonevilla hermitianrankmetriccodesandduality AT ferruhozbudak hermitianrankmetriccodesandduality |