Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866910632777875456 |
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| author | Gao, Yuan Fang, Weijun Xu, Jingke Wang, Dong Hu, Sihuang |
| author_facet | Gao, Yuan Fang, Weijun Xu, Jingke Wang, Dong Hu, Sihuang |
| contents | Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, δ)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, δ)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04323 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes Gao, Yuan Fang, Weijun Xu, Jingke Wang, Dong Hu, Sihuang Information Theory 94B60, 11T71 Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, δ)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, δ)$-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes. |
| title | Optimal $(2,δ)$ Locally Repairable Codes via Punctured Simplex Codes |
| topic | Information Theory 94B60, 11T71 |
| url | https://arxiv.org/abs/2307.04323 |