多集组合批处理码
Multiset Combinatorial Batch Codes
摘要：
由Ishai、Kushilevitz、Ostrovsky和Sahai首先引入的批处理代码模仿$m$服务器上的一组$n$数据项的分布式存储，以这样的方式，可以通过从每个服务器读取至多一些$t$符号来检索任何批次的$k$数据项。组合批处理代码是基于复制的批处理代码，其中每个服务器存储数据项的子集。在本文中，我们提出了一个概括的组合批处理代码，称为多集组合批处理代码（MCBC），其中$n$数据项存储在$m$服务器，这样，任何多集请求的$k$项目，其中任何项目被要求在最多的$r$次，可以通过读取最多的$t$项目从每个服务器检索。这个新的代码族的建立是由最近的工作代码，使高可用性和并行读取分布式存储系统的动机。在此范例下的主要问题是最小化存储在服务器中的项目的数量，给定$n,m,k,r,t$的值，其由$N(n,k,m,t;r)$表示。我们首先给出了MCBC存在的一个充分必要条件。然后，我们给出了$N(n,k,m,t;r)$的几个界和MCBC的构造。特别地，我们确定任何$n\geq \left\lfloor\frac{k-1}{r}\right\rfloor{m\choose k-1}-(m-k+1)A(m,4,k-2)$的$N(n,k,m,1;r)$的值，其中$A(m,4,k-2)$是长度为$m$、距离为4且权重为$k-2$的二进制常重码的最大大小。我们还确定了当$N(n,k,m,1;r)$或$k=m$。
英文摘要：
Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai, mimic a distributed storage of a set of $n$ data items on $m$ servers, in such a way that any batch of $k$ data items can be retrieved by reading at most some $t$ symbols from each server. Combinatorial batch codes, are replication-based batch codes in which each server stores a subset of the data items. In this paper, we propose a generalization of combinatorial batch codes, called multiset combinatorial batch codes (MCBC), in which $n$ data items are stored in $m$ servers, such that any multiset request of $k$ items, where any item is requested at most $r$ times, can be retrieved by reading at most $t$ items from each server. The setup of this new family of codes is motivated by recent work on codes which enable high availability and parallel reads in distributed storage systems. The main problem under this paradigm is to minimize the number of items stored in the servers, given the values of $n,m,k,r,t$, which is denoted by $N(n,k,m,t;r)$. We first give a necessary and sufficient condition for the existence of MCBCs. Then, we present several bounds on $N(n,k,m,t;r)$ and constructions of MCBCs. In particular, we determine the value of $N(n,k,m,1;r)$ for any $n\geq \left\lfloor\frac{k-1}{r}\right\rfloor{m\choose k-1}-(m-k+1)A(m,4,k-2)$, where $A(m,4,k-2)$ is the maximum size of a binary constant weight code of length $m$, distance four and weight $k-2$. We also determine the exact value of $N(n,k,m,1;r)$ when $r\in\{k,k-1\}$ or $k=m$.
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