Almost Optimal Construction of Functional Batch Codes Using Hadamard Codes
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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≤ 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 + ⌊ 2^s/2/√(24)⌋ requests. This paper reduces this gap and shows the existence of codes for k=⌊2/32^s-1⌋ requests with n=2^s-1 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. minimum number of servers for functional k-batch codes. These constructions are mainly based on Hadamard codes and equivalently provide constructions for parallel Random I/O (RIO) codes.
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