AI 中文总结
该研究针对三层网络的向量线性函数计算问题，提出支持约束行空间框架，推导线性计算码存在条件与最小通信负载，将其应用于循环网络MDS目标并给出通用线性构造。
AI 中文摘要
我们研究三层网络中针对固定目标函数和固定源访问模式的向量线性函数计算问题。提出一种支持约束行空间框架，该框架通过全局行空间表示线性计算码，此空间必须包含目标行空间，且由满足网络局部支持约束的行生成。我们证明该表示与线性计算码的存在性等价；对于任意指定的全局行空间，给出其实现的充要条件，并确定中间节点的最小均匀通信负载，该条件基于源访问集上支撑的局部子空间的秩表达，将精确局部实现问题与设计全局行空间的外部问题分离，得到线性计算容量的变分表征。随后将该框架应用于循环网络上的MDS目标，明确仅靠目标行空间足够的情况及需辅助行的情况，在割集界为整数时确定稠密区与稀疏区的容量，对其余稀疏参数，给出可达速率等于割集界整数部分的通用线性构造。
英文摘要
We study vector-linear function computation over three-layer networks with a fixed target function and a fixed source-access pattern. We develop a support-constrained row-space framework that represents a linear computing code by a global row space. This space must contain the target row space and be generated by rows satisfying the local support-constraints of the network. We prove that this representation is equivalent to the existence of a linear computing code. For any prescribed global row space, we give a necessary and sufficient condition for its realization and determine the minimum uniform communication load at the middle nodes. The condition is expressed in terms of the ranks of the local subspaces supported on the source-access sets. It separates the exact local realization problem from the outer problem of designing the global row space and yields a variational characterization of the linear computing capacity. We then apply the framework to MDS targets over cyclic networks. We identify when the target row space alone is sufficient and when auxiliary rows are required. We determine the capacity in the dense regime and in the sparse regime whenever the cut-set bound is integral. For the remaining sparse parameters, we give a general linear construction whose achievable rate equals the integer part of the cut-set bound.
