Almost Optimal Construction of Functional Batch Codes Using Extended Simplex 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=2s-1 requests the optimal solution requires 2s-1 servers. This conjecture is verified for s ≤ 5 but previous work could only show that codes with n=2s-1 servers can support a solution for k=2s-2 + 2s-4 + ⌊2s/2/24⌋ requests. This paper reduces this gap and shows the existence of codes for k=⌊5/6 2s-1⌋ - s requests with the same number of servers. Another construction in the paper provides a code with n=2s+1-2 servers and k=2s requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes.