A \textit{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 $s\leq 5$ 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 Hadamard codes and equivalently provide constructions for \textit{parallel Random I/O (RIO)} codes.
翻译:\ textit{ 功能 $k$- batch} 维度代码 $3 美元由存储线性组合的 线性服务器 $2=2 ⁇ s-1} 信息位数( 或请求) 的多个要求, 都可以用服务器的 $k$ 来回收 线性组合( 或请求) 的 美元 。 这个范例的目标是找到给定值 $ { 2 ⁇ s/2\\ qrt{ 24}\ right\rstom$ 请求的服务器最低数量 。 本文缩小了这一差距, 并显示了 $@l=1$1 线性 5\\\\\\\\\ } 5} 5美元等值服务器的代码的存在, 但先前的工作只能显示 $n=2 ⁇ s-1 服务器的代码可以支持 $k=2 ⁇ s-2} +\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\