Optimal locally repairable codes of distance 3 and 4 via cyclic codes
Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes...
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| Format: | Journal Article |
| Language: | English |
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2020
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| Online Access: | https://hdl.handle.net/10356/143233 |
| _version_ | 1811684718327889920 |
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| author | Luo, Yuan Xing, Chaoping Yuan, Chen |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Luo, Yuan Xing, Chaoping Yuan, Chen |
| author_sort | Luo, Yuan |
| collection | NTU |
| description | Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given by Tamo and Barg are upper bounded by the code alphabet size q. Recently, it was proved through the extension of construction by Tamo and Barg that the length of q-ary optimal locally repairable codes can be q +1 by Jin et al. Surprisingly, Barg et al. presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q 2 . Very recently, it was further shown in the work of Li et al. that there exist q-ary optimal locally repairable codes with the length bigger than q+1 and the distance proportional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q+1. In this paper, we construct a class of optimal locally repairable codes of distances 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen. |
| first_indexed | 2024-10-01T04:33:05Z |
| format | Journal Article |
| id | ntu-10356/143233 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T04:33:05Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | ntu-10356/1432332023-02-28T19:48:39Z Optimal locally repairable codes of distance 3 and 4 via cyclic codes Luo, Yuan Xing, Chaoping Yuan, Chen School of Physical and Mathematical Sciences Science::Mathematics Error Correction Codes Locally Repairable Codes and The Singleton Bound Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given by Tamo and Barg are upper bounded by the code alphabet size q. Recently, it was proved through the extension of construction by Tamo and Barg that the length of q-ary optimal locally repairable codes can be q +1 by Jin et al. Surprisingly, Barg et al. presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q 2 . Very recently, it was further shown in the work of Li et al. that there exist q-ary optimal locally repairable codes with the length bigger than q+1 and the distance proportional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q+1. In this paper, we construct a class of optimal locally repairable codes of distances 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen. Ministry of Education (MOE) Accepted version Y. Luo was supported by the National Natural Science Foundation of China under Grant 61571293. C. Xing was supported by the Singapore MOE Tier 1 under Grant RG25/16. C. Yuan was supported by ERC H2020 under Grant 74079 (ALGSTRONGCRYPTO). 2020-08-14T04:13:01Z 2020-08-14T04:13:01Z 2018 Journal Article Luo, Y., Xing, C., & Yuan, C. (2019). Optimal locally repairable codes of distance 3 and 4 via cyclic codes. IEEE Transactions on Information Theory, 65(2), 1048-1053. doi:10.1109/TIT.2018.2854717 0018-9448 https://hdl.handle.net/10356/143233 10.1109/TIT.2018.2854717 2-s2.0-85049788772 2 65 1048 1053 en IEEE Transactions on Information Theory © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: https://doi.org/10.1109/TIT.2018.2854717. application/pdf |
| spellingShingle | Science::Mathematics Error Correction Codes Locally Repairable Codes and The Singleton Bound Luo, Yuan Xing, Chaoping Yuan, Chen Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title | Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title_full | Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title_fullStr | Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title_full_unstemmed | Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title_short | Optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| title_sort | optimal locally repairable codes of distance 3 and 4 via cyclic codes |
| topic | Science::Mathematics Error Correction Codes Locally Repairable Codes and The Singleton Bound |
| url | https://hdl.handle.net/10356/143233 |
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