Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six
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2025
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| author | Alshuhail, Altaf |
| author_facet | Alshuhail, Altaf |
| contents | We consider codes over the two semi-local non-unital rings of order six, \[ H_{23} = \langle a,b \mid 2a=0, 3b = 0, a^2=a, b^2 = 0, ab = 0 = ba \rangle,\] and \[H_{32} = \langle a,b \mid 2a=0, 3b = 0, a^2=0, b^2 = b, ab = 0 = ba \rangle. \] with respect to a symplectic inner product. Via the decomposition $\mathcal{C} = a\,\mathcal{C}_a\oplus b\,\mathcal{C}_b$, any $H_z$-code splits into a binary component $\mathcal{C}_a$ and a ternary component $\mathcal{C}_b$; this yields characterizations of symplectic self-orthogonal, self-dual, and quasi self-dual (QSD) codes, and allows the introduction of symplectic nice and symplectic linear complementary dual (LCD) codes. Using the automorphism groups of $\mathcal{C}_a$ and $\mathcal{C}_b$ and double-coset enumeration, we classify, up to permutation equivalence, all symplectic self-orthogonal and QSD $H_z$-codes of short even lengths. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six Alshuhail, Altaf Rings and Algebras Group Theory Primary 94 B05, Secondary 16 A10 We consider codes over the two semi-local non-unital rings of order six, \[ H_{23} = \langle a,b \mid 2a=0, 3b = 0, a^2=a, b^2 = 0, ab = 0 = ba \rangle,\] and \[H_{32} = \langle a,b \mid 2a=0, 3b = 0, a^2=0, b^2 = b, ab = 0 = ba \rangle. \] with respect to a symplectic inner product. Via the decomposition $\mathcal{C} = a\,\mathcal{C}_a\oplus b\,\mathcal{C}_b$, any $H_z$-code splits into a binary component $\mathcal{C}_a$ and a ternary component $\mathcal{C}_b$; this yields characterizations of symplectic self-orthogonal, self-dual, and quasi self-dual (QSD) codes, and allows the introduction of symplectic nice and symplectic linear complementary dual (LCD) codes. Using the automorphism groups of $\mathcal{C}_a$ and $\mathcal{C}_b$ and double-coset enumeration, we classify, up to permutation equivalence, all symplectic self-orthogonal and QSD $H_z$-codes of short even lengths. |
| title | Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six |
| topic | Rings and Algebras Group Theory Primary 94 B05, Secondary 16 A10 |
| url | https://arxiv.org/abs/2508.16344 |