Almost Optimal Construction of Functional Batch Codes Using Hadamard Codes

A functional k-batch code of dimension s consists of n servers storing linear combinations of s linearly independent information bits. Any multiset request of size k of linear combinations (or requests) of the information bits can be recovered by k disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of s and k. A recent conjecture states that for any k=2^{s-1} requests the optimal solution requires 2^{s}-1 servers. This conjecture is verified for s\leqslant 5 but previous work could only show that codes with n=2^{s}-1 servers can support a solution for k=2^{s-2}+2^{s-4}+ \left\lfloor\frac{2^{s/2{\sqrt{24 \right\rfloor requests. This paper reduces this gap and shows the existence of codes for k= \lfloor\frac{2}{3}2^{s-1}\rfloor requests with the same number of servers. Another construction in the paper provides a code with n=2^{s+1}-2 servers and k=2^{s} requests, which is an optimal result. These constructions are mainly based on Hadamard codes and equivalently provide constructions for parallel Random I/O (RIO) codes.