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Almost Optimal Construction of Functional Batch Codes Using Extended Simplex Codes

机译：Almost Optimal Construction of Functional Batch Codes Using Extended Simplex 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 $s leqslant 5$ 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 $k=lfloor frac {5}{6}2^{s-1} rfloor - s$ 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 extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes.

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