Optimal Functional $2^{s-1}$-Batch Codes: Exploring New Sufficient Conditions

TL;DR

This paper investigates the structure of optimal functional $2^{s-1}$-batch codes, establishing equivalences with other problems and deriving new sufficient conditions to better understand and identify optimal solutions.

Contribution

It introduces new sufficient conditions for optimal functional batch codes and demonstrates their equivalence to other problems, advancing the theoretical understanding of these codes.

Findings

Derived new sufficient conditions for optimal codes

Established equivalences with related problems

Enhanced methods for identifying optimal solutions

Abstract

A functional $k$-batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. These codes are designed to recover any multiset of $k$ requests, each being a linear combination of the information bits, by $k$ disjoint subsets of servers. A recent conjecture suggests that for any set of $k = 2^{s-1}$ requests, the optimal solution requires $2^s-1$ servers. This paper shows that the problem of functional $k$-batch codes is equivalent to several other problems. Using these equivalences, we derive sufficient conditions that improve understanding of the problem and enhance the ability to find the optimal solution.