Linear network codes for vector-linear network function computation over three-layer networks

2026-08-02

AI Summary

This work addresses vector linear network function computation in three-layer networks under a fixed target function and given source node access patterns. It introduces a “support-constrained row space” framework that models linear computation codes as global row spaces containing the target row space while satisfying local support constraints, thereby decoupling local implementation from global design. Within this framework, the existence of linear computation codes is characterized via an equivalent row space condition, yielding necessary and sufficient conditions for feasibility, a lower bound on communication load, and a variational expression for linear computing capacity. By integrating row space analysis, rank conditions, MDS code constructions, and cut-set bounds, the study precisely characterizes computing capacity in cyclic networks for both dense and sparse regimes and provides a general linear construction achieving the integer part of the cut-set bound for arbitrary sparsity parameters.

Abstract

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.

Problem

• network function computation
• linear network codes
• three-layer networks
• vector-linear
• computing capacity

Innovation

• support-constrained row space
• linear network function computation
• three-layer networks
• MDS targets
• cut-set bound