Combinatorial Batch Codes: A Lower Bound and Optimal Constructions

TL;DR

This paper establishes a lower bound and provides explicit constructions for optimal combinatorial batch codes, advancing the understanding of minimal storage solutions for data retrieval across multiple servers.

Contribution

It introduces a lower bound on total storage for CBCs and offers explicit constructions for optimal and near-optimal CBCs, partly resolving an open problem.

Findings

Derived a lower bound on total storage for CBCs

Constructed optimal CBCs for certain parameter ranges

Provided almost optimal CBCs for other parameter ranges

Abstract

Batch codes, introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [1], 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 [2], Paterson, Stinson and Wei 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 [2].