组合批码的去随机化构造
Derandomized Construction of Combinatorial Batch Codes
摘要：
组合批处理代码（CBCs），由Ishai等人在STOC 2004中引入的基于复制的批处理代码变体，抽象了以下数据分布问题：$n$数据项将以这样的方式在$m$服务器之间被复制，使得$n$数据项的任何$k$可以通过从每个服务器读取至多一个项来检索，其中$m$服务器上的总存储量被限制为$N$。给定参数$m, c,$和$k$，其中$c$和$k$是常数，挑战性问题之一是构造$c$ -均匀CBC（CBC，其中每个数据项在精确的$c$服务器之间复制），其最大化$n$的值。在这项工作中，我们提出了明确的建设$c$ -均匀CBCs与$Ω(m^{c-1+{1 \over k}})$数据项。该结构具有服务器几乎规则的特性，即，存储在每个服务器中的数据项的数量在$[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$范围内。该结构是通过更好的分析和去随机化Ishai等人提出的随机结构。分析揭示了几乎规则的服务器，这方面迄今尚未在文献中得到解决。去随机化导致针对$c$的大范围值的显式构造（对于给定的$m$和$k$），其中没有具有类似参数的其他显式构造，即，用$n = Ω(m^{c-1+{1 \over k}})$，是已知的。最后，我们讨论了并行去随机化的可能性。
英文摘要：
Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in STOC 2004, abstracts the following data distribution problem: $n$ data items are to be replicated among $m$ servers in such a way that any $k$ of the $n$ data items can be retrieved by reading at most one item from each server with the total amount of storage over $m$ servers restricted to $N$. Given parameters $m, c,$ and $k$, where $c$ and $k$ are constants, one of the challenging problems is to construct $c$-uniform CBCs (CBCs where each data item is replicated among exactly $c$ servers) which maximizes the value of $n$. In this work, we present explicit construction of $c$-uniform CBCs with $Ω(m^{c-1+{1 \over k}})$ data items. The construction has the property that the servers are almost regular, i.e., number of data items stored in each server is in the range $[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$. The construction is obtained through better analysis and derandomization of the randomized construction presented by Ishai et al. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of $c$ (for given $m$ and $k$) where no other explicit construction with similar parameters, i.e., with $n = Ω(m^{c-1+{1 \over k}})$, is known. Finally, we discuss possibility of parallel derandomization of the construction.
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