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- W2806362959 abstract "Locally repairable codes with locality $r$ ($r$-LRCs for short) were introduced by Gopalan et al. cite{1} to recover a failed node of the code from at most other $r$ available nodes. And then $(r,delta)$ locally repairable codes ($(r,delta)$-LRCs for short) were produced by Prakash et al. cite{2} for tolerating multiple failed nodes. An $r$-LRC can be viewed as an $(r,2)$-LRC. An $(r,delta)$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct $q$-ary optimal $(r,delta)$-LRCs with length much larger than $q$. Surprisingly, Luo et al. cite{3} presented a construction of $q$-ary optimal $r$-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of $q$) via cyclic codes. In this paper, inspired by the work of cite{3}, we firstly construct two classes of optimal cyclic $(r,delta)$-LRCs with unbounded lengths and minimum distances $delta+1$ or $delta+2$, which generalize the results about the $delta=2$ case given in cite{3}. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic $(r,delta)$-LRCs with unbounded length and larger minimum distance $2delta$. Furthermore, when $delta=3$, we give another class of optimal cyclic $(r,3)$-LRCs with unbounded length and minimum distance $6$." @default.
- W2806362959 created "2018-06-13" @default.
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- W2806362959 date "2018-05-31" @default.
- W2806362959 modified "2023-10-14" @default.
- W2806362959 title "Optimal cyclic $(r,delta)$ locally repairable codes with unbounded length" @default.
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- W2806362959 doi "https://doi.org/10.48550/arxiv.1805.12345" @default.
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