Combinatorial Batch Codes: A Lower Bound and Optimal Constructions

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].

💡 Research Summary

This paper studies combinatorial batch codes (CBCs) with the parameter t = 1, i.e., each server may be accessed at most once when retrieving any k data items. An (n, N, k, m)‑CBC stores n items (possibly with replication) across m servers, using a total of N stored copies, and must guarantee that any k distinct items can be recovered by reading at most one copy from each server. The central goal is to determine the minimal possible total storage N(n, k, m) for given n, k, m and to construct explicit CBCs that achieve this bound.

The authors first model a CBC as a set system (S, X) where S is the set of servers and X = {X₁,…,Xₙ} is a family of subsets of S, each Xᵢ indicating the servers that hold a copy of item i. The retrieval requirement translates directly into Hall’s theorem: for any collection of r ≤ k items, the union of their server sets must contain at least r distinct servers. This yields two equivalent restricted Hall conditions, denoted HC₁