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Error Decoding of Locally Repairable and Partial MDS Codes

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 نشر من قبل Lukas Holzbaur
 تاريخ النشر 2019
  مجال البحث الهندسة المعلوماتية
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In this work it is shown that locally repairable codes (LRCs) can be list-decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error-correction capabilities. The corresponding decoding radius is derived and the asymptotic behavior is analyzed. A general list-decoding algorithm for LRCs that achieves this radius is proposed along with an explicit realization for LRCs that are subcodes of Reed--Solomon codes (such as, e.g., Tamo--Barg LRCs). Further, a probabilistic algorithm of low complexity for unique decoding of LRCs is given and its success probability is analyzed. The second part of this work considers error decoding of LRCs and partial maximum distance separable (PMDS) codes through interleaved decoding. For a specific class of LRCs the success probability of interleaved decoding is investigated. For PMDS codes, it is shown that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance such that the probability of successful decoding approaches $1$ when the code length goes to infinity.



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We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance with the probability of successful decoding approaching $1$, when the code length goes to infinity.
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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 $6$.
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