Edge-Subset Lattice and Its Application to Linear Network Error Correction Coding

In this paper, we first explore the underlying mathematical structure of edge subsets on a finite directed acyclic graph in using a lattice-theoretic approach. We prove that a collection of edge subsets, depending on different conditions it satisfies, together with the corresponding “cut-separating” partial order, can form a meet-semilattice, a join-semilattice or a lattice. The bottom and top thus derived generalize the concept of the primary minimum cut introduced by Guang and Yeung (2018) and hence we provide a new way from the lattice-theoretic point of view to understand the primary minimum cut and its existence and uniqueness. The introduced concepts and obtained results regarded as a bridge connect graph theory and lattice theory, which appear to be of fundamental interest in graph theory, lattice theory, and even beyond. In addition, we consider linear network error correction (LNEC) coding when errors may occur on edges of a communication network of which the topology is known. In LNEC coding, the minimum required field size for the existence of LNEC codes, in particular LNEC maximum distance separable (MDS) codes which are regarded as a most important type of optimal codes, is an open problem not only of theoretical interest but also of practical importance. By applying the approach of the above edge-subset (semi-)lattice, we obtain an improved upper bound on the minimum required field size for the existence of LNEC (MDS) codes. We quantify the improvement over the existing results by both theoretical analysis and numerical simulations and thus show that the improvement is in general significant. The improved upper bound, which is graph-theoretic, depends only on the network topology and requirement of the error correction capability but not on a specific code construction. We also develop an efficient algorithm that can compute the bound in a linear time of the number of edges.