On Metrics for Error Correction in Network Coding

TL;DR

This paper introduces new metrics for error correction in network coding, providing bounds, constructions, and decoding strategies for both coherent and noncoherent scenarios under adversarial models.

Contribution

It defines the injection metric for noncoherent network coding and demonstrates its advantages over existing metrics, along with constructions achieving the Singleton bound.

Findings

Universal network error correcting codes achieving the Singleton bound can be constructed.

The injection metric allows for correcting more errors than the subspace metric in non-constant-dimension codes.

The approach to adversarial error correction is general and applicable beyond network coding.

Abstract

The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the injection metric, which is closely related to, but different than, the subspace metric of K\"otter and Kschischang. In particular, in the case of a non-constant-dimension code, the decoder associated with the injection metric is shown to correct more errors then a minimum-subspace-distance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding.