Title: Multiset Combinatorial Batch Codes,
Abstract: 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 $k=m$.