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### 两类最优循环局部修复码的构造

1. 合肥工业大学数学学院, 合肥 230601
• 收稿日期:2021-05-20 修回日期:2021-09-24 出版日期:2022-02-25 发布日期:2022-03-21
• 基金资助:
国家自然科学基金（61972126，61772168，61802102）资助课题.

CHEN Min, KAI Xiaoshan. Constructions of Two Classes of Optimal Cyclic Locally Repairable Codes[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(2): 487-494.

### Constructions of Two Classes of Optimal Cyclic Locally Repairable Codes

CHEN Min, KAI Xiaoshan

1. School of Mathematics, Hefei University of Technology, Hefei 230601
• Received:2021-05-20 Revised:2021-09-24 Online:2022-02-25 Published:2022-03-21

Locally repairable codes are a class of erasure codes which can repair multiple failed nodes. They are widely used in the distributed storage systems. A main topic in the distributed storage coding is to construct optimal locally repairable codes at present. In this paper, the following two classes of optimal ${(r,\delta)}$ locally repairable codes based on cyclic codes over $\mathbb{F}_{q}$ are constructed:1) $[3(q+1),3(q+1)-3\delta+1,\delta+2]$, where $q\equiv1(\bmod~6)$, $r+\delta-1=q+1$ and $2\leq\delta\leq q-1$ is even; 2) $[3(q-1),3(q-1)-3\delta+2,\delta+1]$, where $q\equiv7(\bmod~9)$, $r+\delta-1=q-1$, $2\leq\delta\leq\frac{2(q-1)}{3}$ is even with $\delta\not\equiv0(\bmod~6)$.

MR(2010)主题分类:

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 [1] Gopalan P, Huang C, Simitci H, et al. On the locality of codeword symbols. IEEE Transactions on Information Theory, 2012, 58(11):6925-6934.[2] Prakash N, Kamath G M, Lalitha V, et al. Optimal linear codes with a local-error-correction property. Proceedings of IEEE International Symposium on Information Theory, Cambridge, MA, 2012, 2776-2780.[3] Hao J, Xia S T, Chen B. On optimal ternary locally repairable codes. Proceedings of IEEE International Symposium on Information Theory, Aachen, Germany, 2017, 171-175.[4] Krishnan M N, Puranik B, Kumar P V, et al. Exploiting locality for improved decoding of binary cyclic codes. IEEE Transactions on Communications, 2018, 66(6):2346-2358.[5] Jin L F. Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes. IEEE Transactions on Information Theory, 2019, 65(8):4658-4663.[6] Cai H, Miao Y, Schwartz M, et al. On optimal locally repairable codes with super-linear length. IEEE Transactions on Information Theory, 2020, 66(8):4853-4868.[7] Goparaju S, Calderbank R. Binary cyclic codes that are locally repairable. Proceedings of IEEE International Symposium on Information Theory, Honolulu, HI, 2014, 676-680.[8] Zeh A, Yaakobid E. Optimal linear and cyclic locally repairable codes over small fields. IEEE Information Theory Workshop, Jerusalem, Israel, 2015.[9] Tamo I, Barg A, Goparaju S, et al. Cyclic LRC codes and their subfield subcodes. Proceedings of IEEE International Symposium on Information Theory, Hong Kong, 2015, 126-1266.[10] Chen B, Xia S T, Hao J, et al. Constructions of optimal cyclic (r,δ) locally repairable codes. IEEE Transactions on Information Theory, 2018, 64(4):2499-2511.[11] Luo Y, Xing C P, Yuan C. Optimal locally repairable codes of distance 3 and 4 via cyclic codes. IEEE Transactions on Information Theory, 2019, 65(2):1048-1053.[12] Sun Z H, Zhu S X, Wang L Q. Optimal constacyclic locally repairable codes. IEEE Communications Letters, 2019, 23(2):206-209.[13] Fang W J, Fu F W. Optimal cyclic (r,δ) locally repairable codes with unbounded length. Finite Fields and Their Applications, 2020, 63(3):101650(1-14).[14] Tan P, Zhou Z C, Yan H D, et al. Optimal cyclic locally repairable codes via cyclotomic polynomials. IEEE Communications Letters, 2019, 23(2):202-205.[15] Qiu J, Zheng D B, Fu F W. New constructions of optimal cyclic (r, δ) locally repairable codes from their zeros. IEEE Transactions on Information Theory, 2021, 67(3):1596-1608.[16] Qian J F, Zhang L N. New optimal cyclic locally recoverable codes of length $n=2(q +1)$. IEEE Transactions on Information Theory, 2020, 66(1):233-239.[17] MacWilliams F, Sloane N. The Theory of Error-Correcting Codes. Amsterdam:North-Holland, 1981.[18] Huffman W, Pless V. Fundamentals Error-Correcting Codes. Cambridge:Cambridge University Press, 2003.
 [1] 胡建，曹喜望. 自共轭互反多项式的推广[J]. 系统科学与数学, 2020, 40(8): 1507-1516. [2] 王小强，曾丽琦，刘丽娟. 几类循环码的重量分布研究[J]. 系统科学与数学, 2018, 38(4): 484-496. [3] 袁文红. 伽罗瓦内积下LCD常循环码和LCD MDS码的研究[J]. 系统科学与数学, 2018, 38(12): 1464-1476. [4] 于龙，胡鹏，刘修生，刘宏伟. 一类循环码的完全重量分布[J]. 系统科学与数学, 2018, 38(10): 1206-1212. [5] 李兰强，刘丽. 环${Z}_{ 4}+{u}{Z}_{ 4}$上一类重根常循环码[J]. 系统科学与数学, 2017, 37(3): 870-881. [6] 郑大彬,刘犇，王小强. 一类具有三个非零点的循环码的重量分布[J]. 系统科学与数学, 2016, 36(4): 573-590. [7] 黄炎，开晓山，宛金龙.  环$\mathbb{Z}_{p^2}$上的循环自正交码的构造[J]. 系统科学与数学, 2016, 36(12): 2473-2480. [8] 高莹，梅佳. 两类基于完全非线性函数的线性码[J]. 系统科学与数学, 2014, 34(2): 129-134. [9] 张元婷，朱士信，王立启. 环${\mathbb{F}}_{{\bm p}^{\bm m}}+{\bm u}\mathbb{F}_{{\bm p}^{\bm m}}+{\bm v}\mathbb{F}_{{\bm p}^{\bm m}}+{\bm u \bm v}\mathbb{F}_{{\bm p}^{\bm m}}$上的一类重根常循环码[J]. 系统科学与数学, 2014, 34(2): 135-144. [10] 开晓山;朱士信. 环GR(4,2)上一类负循环码的Gray象[J]. 系统科学与数学, 2010, 30(3): 334-340. [11] 王维琼;张文鹏. 关于设计距离为7的BCH码的非循环等价类[J]. 系统科学与数学, 2006, 26(1): 42-047.