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A Class of Binary Locally Repairable Codes

机译：一类可本地修复的二进制代码

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摘要

An $(n,k)$ erasure code that can recover any coded symbol by at most $r$ other coded symbols is called a locally repairable code (LRC) with locality $r$ . LRCs have been recently implemented in distributed storage systems. Coding complexity reduction can be significantly decreased by using binary LRCs (BLRCs) as they eliminate costly multiplication calculation. In this paper, motivated by the recently erasure codes with $d=4$ used in practice, we propose BLRCs when $(r+1)mid n$ and $d=4$ . We prove that our proposed binary codes are optimal for $rin {1,3}$ , meaning that neither their locality nor their minimum distance can be improved by non-binary codes. For $rgeq 4$ , our proposed binary codes offer near-optimal code rate, with a rate gap of $mathcal {O}({log r}/{n})$ compared with optimal non-binary codes. While keeping the bulk of code structure binary, we eliminate this rate gap by using fields with sizes as small as $r+2$ for only two redundant symbols. These non-binary codes still eliminate the need for costly multiplications in many operations including a single failure repair (a dominant repair scenario). Using the construction of spanning BLRC with $d=4$ as a back- one, we also construct LRCs with minimum distance $dgeq 6$ . Furthermore, we obtain a closed-form equation for the mean-time to data-loss of arbitrary erasure codes.
机译：可以通过最多$ r $个其他编码符号恢复任何编码符号的$（n，k）$擦除码称为局部性$ r $的本地可修复码（LRC）。 LRC最近已在分布式存储系统中实现。通过使用二进制LRC（BLRC），可以减少编码复杂度，因为它们消除了昂贵的乘法计算。在本文中，受最近在实践中使用$ d = 4 $的擦除代码的启发，我们提出了$（r + 1）mid n $和$ d = 4 $时的BLRC。我们证明了我们提出的二进制代码对于$ rin {1,3} $是最佳的，这意味着非二进制代码无法改善其位置或最小距离。对于$ rgeq 4 $，我们提出的二进制代码提供了接近最佳的代码速率，与最优非二进制代码相比，速率差距为$ mathcal {O}（{log r} / {n}）$。在保持大部分代码结构为二进制的同时，我们通过对两个冗余符号使用大小仅为$ r + 2 $的字段来消除此速率差距。这些非二进制代码仍然消除了包括单次故障修复（主要修复方案）在内的许多操作中昂贵的乘法运算的需要。使用以$ d = 4 $作为跨度的BLRC的构造，我们还以最小距离$ dgeq 6 $构造LRC。此外，对于任意擦除码的数据丢失的平均时间，我们获得了一个封闭式方程。

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