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

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.