Linear Network Error Correction Coding Revisited

We consider linear network error correction (LNEC) coding when errors may occur on the edges of a communication network of which the topology is known. In this paper, we first revisit and explore the framework of LNEC coding, and then unify two well-known LNEC coding approaches. In LNEC coding, LNEC maximum distance separable (MDS) codes are a type of most important optimal codes. However, the minimum required field size for the existence of LNEC MDS codes is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we obtain an improved lower bound on the required field size by developing a graphtheoretic approach. The improvement over the existing results is in general significant. Furthermore, by applying the graphtheoretic approach to the framework of LNEC coding, we obtain a significantly enhanced characterization of the capability of an LNEC code in terms of its minimum distance.