On Erasure Combinatorial Batch Codes

TL;DR

This paper studies erasure combinatorial batch codes with a focus on minimal total storage, establishing bounds, optimal configurations, and connections to graph theory for parameters where each server reads only one item.

Contribution

It determines the minimum total storage for certain parameters, relates optimal codes to maximum packings, and links bounds to graph girth conditions.

Findings

Optimal total storage for specific parameter ranges

Connection between erasure batch codes and maximum packings

Lower bounds related to graph girth conditions

Abstract

Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers, each of which stores a subset of the items. A batch code is an arrangement for storing items on servers so that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Silberstein defined an erasure batch code (with redundancy $r$) as a batch code in which any $k$ items can be retrieved by reading at most $t$ items from each server, while any $r$ servers are unavailable (failed). In this paper, we investigate erasure batch codes with $t=1$ (each server can read at most one item) in a combinatorial manner. We determine the optimal (minimum) total storage of an erasure batch code for several ranges of parameters. Additionally, we relate optimal erasure batch codes to maximum packings. We also identify a necessary lower bound for the total storage of an erasure batch code, and we relate parameters for which this trivial lower bound is achieved to the existence of graphs with appropriate girth.