Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes
Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes
复制标题
容量接近代码的更新效率和本地可修复性限制
DOI：
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发表时间：
2013
影响因子：
16.4
通讯作者：
G. Wornell
中科院分区：
文献类型：
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作者：
A. Mazumdar;V. Chandar;G. Wornell
受分布式存储应用的驱动，我们研究当单个信息符号改变时，容量可达码能够被有效更新的程度，以及当单个编码符号丢失时，此类编码能够被有效修复的程度。具体而言，我们首先推导出能够实现最优纠错和更新效率的条件。我们确定，如果要在二进制删除信道或二进制对称信道上以趋近于零的错误概率实现任何非平凡的码率，那么因单个信息比特的改变而应改变的编码比特数必须与码的分组长度成对数关系。此外，我们表明存在具有这种缩放关系的容量可达码。关于局部可修复性，我们对恢复单个丢失编码比特所需的剩余编码比特数给出了紧的上界和下界。特别地，我们表明当一个最优码的码率比容量低\(\varepsilon\)时，恢复一个丢失符号所需的码字符号的最大数量必须与\(\log\frac{1}{\varepsilon}\)成比例。我们还对这些结果进行了若干变体和扩展，包括对率失真编码问题的研究。
Motivated by distributed storage applications, we investigate the degree to which capacity achieving codes can be efficiently updated when a single information symbol changes, and the degree to which such codes can be efficiently repaired when a single encoded symbol is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible. We establish that the number of encoded bits that should change in response to a change in a single information bit must scale logarithmically in the block-length of the code, if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show that there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost encoded bit. In particular, we show that when the rate of an optimal code is ε below capacity, the maximum number of codeword symbols required to recover one lost symbol must scale as log1/ε. Several variations on-and extensions of-these results are also developed, including to the problem of rate-distortion coding.