论文标题
关于超值储存率超过图形的速率
On Extremal Rates of Secure Storage over Graphs
论文作者
论文摘要
安全的存储代码映射$ k $源符号，$ l_w $位的每个符号，到$ n $编码符号，$ l_v $ bits的每个符号，因此每个编码符号都存储在图的节点中。该图的每个边缘都与$ k $源符号的$ d $关联，以便从边缘连接的一对节点中，我们可以解码$ d $源符号，并且不学习有关其余$ k-d $ source符号的信息；或边缘与没有源符号相关联，使得从边缘连接的一对节点中，$ k $源符号都没有透露。比率$ l_w/l_v $称为安全存储代码的符号速率，最高可能的符号速率称为容量。 我们表征了所有图表，当时$ d = 1 $，安全存储代码的容量等于$ 1 $。该结果被推广到$ d> 1 $，即我们表征了所有图表，在轻度条件下，安全存储代码的容量等于$ 1/d $，对于任何节点，与其每个连接边缘相关的源符号都不包含一个共同元素。此外，我们表征了所有图形，安全存储代码的容量等于$ 2/d $。
A secure storage code maps $K$ source symbols, each of $L_w$ bits, to $N$ coded symbols, each of $L_v$ bits, such that each coded symbol is stored in a node of a graph. Each edge of the graph is either associated with $D$ of the $K$ source symbols such that from the pair of nodes connected by the edge, we can decode the $D$ source symbols and learn no information about the remaining $K-D$ source symbols; or the edge is associated with no source symbols such that from the pair of nodes connected by the edge, nothing about the $K$ source symbols is revealed. The ratio $L_w/L_v$ is called the symbol rate of a secure storage code and the highest possible symbol rate is called the capacity. We characterize all graphs over which the capacity of a secure storage code is equal to $1$, when $D = 1$. This result is generalized to $D> 1$, i.e., we characterize all graphs over which the capacity of a secure storage code is equal to $1/D$ under a mild condition that for any node, the source symbols associated with each of its connected edges do not include a common element. Further, we characterize all graphs over which the capacity of a secure storage code is equal to $2/D$.
