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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.14674 |
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| _version_ | 1866912337809637376 |
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| author | Bose, Mrinal Kanti Parampalli, Udaya Singh, Abhay Kumar |
| author_facet | Bose, Mrinal Kanti Parampalli, Udaya Singh, Abhay Kumar |
| contents | Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding \cite{SETA5} from several known classes of permutation monomials and trinomials over $\mathbb{F}_{2^m}$. We present several infinite families of binary cyclic codes of length $2^m-1$ with dimensions larger than $(2^m-1)/2$. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters $[2^m-1,2^m-2-3m,8]$, where $m\geq 5$ is odd, according to the sphere-packing bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Binary cyclic codes from permutation polynomials over $\mathbb{F}_{2^m}$ Bose, Mrinal Kanti Parampalli, Udaya Singh, Abhay Kumar Information Theory 94B15, 11T71, 11T06 B.4.1; H.1.1 Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding \cite{SETA5} from several known classes of permutation monomials and trinomials over $\mathbb{F}_{2^m}$. We present several infinite families of binary cyclic codes of length $2^m-1$ with dimensions larger than $(2^m-1)/2$. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters $[2^m-1,2^m-2-3m,8]$, where $m\geq 5$ is odd, according to the sphere-packing bound. |
| title | Binary cyclic codes from permutation polynomials over $\mathbb{F}_{2^m}$ |
| topic | Information Theory 94B15, 11T71, 11T06 B.4.1; H.1.1 |
| url | https://arxiv.org/abs/2504.14674 |