Some New Optimal Skew Cyclic Codes With Derivation
Our study included a class of cyclic codes named <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }-$ </tex-math></inline-formula>cyclic codes over the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal {R}=\mathbb {F}...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
IEEE
2025-01-01
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| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/11048936/ |
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| Summary: | Our study included a class of cyclic codes named <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }-$ </tex-math></inline-formula>cyclic codes over the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal {R}=\mathbb {F}_{2^{m}} + u\mathbb {F}_{2^{m}}+u^{2}\mathbb {F}_{2^{m}}$ </tex-math></inline-formula>, where m is an odd positive integer with <inline-formula> <tex-math notation="LaTeX">$u^{3}=1$ </tex-math></inline-formula>. These codes are modules over the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal {R}{[x,\alpha,\delta _{\alpha,\zeta }]}$ </tex-math></inline-formula> with automorphism <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-derivation <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }$ </tex-math></inline-formula>(where <inline-formula> <tex-math notation="LaTeX">$\zeta \in \mathcal {R}$ </tex-math></inline-formula>). The structure of ring <inline-formula> <tex-math notation="LaTeX">$\mathcal {R}{[x,\alpha,\delta _{\alpha,\zeta }]}$ </tex-math></inline-formula> is presented to develop linear codes over the ring. The center of the ring is characterized for all inner <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-derivations <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }$ </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">$\zeta $ </tex-math></inline-formula> is taken as an arbitrary element of the ring fixed by <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. In addition, an algorithm was developed and implemented in MAGMA to perform division in this ring. The generator and parity-check matrices for <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }-$ </tex-math></inline-formula>cyclic codes are proposed, and examples of new optimal <inline-formula> <tex-math notation="LaTeX">$\delta _{\alpha,\zeta }-$ </tex-math></inline-formula>cyclic codes with better minimum Hamming distance than those already existing in the Markus Grassl code tables are provided. |
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| ISSN: | 2169-3536 |