On the Minimum Length of Functional Batch Codes with Small Recovery Sets
Kristiina Oksner, Henk D. L. Hollmann, Ago-Erik Riet, Vitaly Skachek
TL;DR
This work addresses the minimum-length problem for linear functional batch codes over ${\mathbb{F}}_2$ with recovery-set size capped by $r$. It introduces a generating-function framework centered on $\theta_{t,r}(n)$ and derives a key recursion $\theta_{t,r}(n)=\sum_{i=1}^r {n \choose i}\theta_{t-1,r}(n-i)$, yielding lower bounds such as $2^k-1 \le (n-(t-1)/2)(n-t)^{r-1}/(r-1)!$ for $n \ge t+r$ and a specialized bound for $r=2$: $n \ge \sqrt{2\,(2^k-1)}+3t/4-5/4$. The paper also discusses exact and numerical evaluations of $\theta_{t,r}(n)$ and compares bounds with double-simplex constructions, highlighting practical design implications for load balancing and private information retrieval. Overall, the results provide concrete theoretical limits and computational insights for functional batch codes under small recovery-set constraints, guiding efficient code design in distributed storage systems.
Abstract
Batch codes are of potential use for load balancing and private information retrieval in distributed data storage systems. Recently, a special case of batch codes, termed functional batch codes, was proposed in the literature. In functional batch codes, users can query linear combinations of the information symbols, and not only the information symbols themselves, as is the case for standard batch codes. In this work, we consider linear functional batch codes with the additional property that every query is answered by using only a small number of coded symbols. We derive bounds on the minimum length of such codes, and evaluate the results by numerical computations.