Synchronization of High-Dimensional Linear Networks over Finite Fields

TL;DR

This paper studies the synchronization of complex high-dimensional linear networks over finite fields, providing new necessary and sufficient conditions and extending previous 1-dimensional results with a novel approach.

Contribution

It generalizes synchronization conditions from 1-dimensional to high-dimensional networks over finite fields using linear transformations and invariant subspaces.

Findings

Derived necessary and sufficient synchronization conditions

Extended results from 1D to high-dimensional networks

Validated theoretical results with numerical examples

Abstract

This paper investigates the synchronization problems for general high-dimensional linear networks over finite fields. By using the technique of linear transformations and invariant subspaces for linear spaces over finite fields, several necessary and sufficient conditions for the synchronization of high-dimensional linear networks over finite fields are proposed. This paper not only generalizes the existing results from 1-dimensional to high-dimensional linear networks but also adopts a new approach. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.