Article Contents
Article Contents

# Combinatorial batch codes: A lower bound and optimal constructions

• Batch codes, introduced by Ishai et al. in [11], are methods for solving the following data storage problem: $n$ data items are to be stored in $m$ servers in such a way that any $k$ of the $n$ items can be retrieved by reading at most $t$ items from each server, and that the total number of items stored in $m$ servers is $N$. A combinatorial batch code (CBC) is a batch code where each data item is stored without change, i.e., each stored data item is a copy of one of the $n$ data items.
One of the basic yet challenging problems is to find optimal CBCs, i.e., CBCs for which total storage ($N$) is minimal for given values of $n$, $m$, $k$, and $t$. In [13], Paterson et al. exclusively studied CBCs and gave constructions of some optimal CBCs.
In this article, we give a lower bound on the total storage ($N$) for CBCs. We give explicit construction of optimal CBCs for a range of values of $n$. For a different range of values of $n$, we give explicit construction of optimal and almost optimal CBCs. Our results partly settle an open problem of [13].
Mathematics Subject Classification: Primary: 05D05, 68P30; Secondary: 68P20.

 Citation:

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