On Metrics for Error Correction in 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 Kötter 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.

A module minimization approach to Gabidulin decoding via interpolation

“…e.g. [22]). The analogy in the classical Hamming metric set-up is the error locator polynomial, whose roots indicate the error locations.…”

A Survey of Linear Network Coding and Network Error Correction Code Constructions and Algorithms

“…Silva and Kschischang also proposed a metric which better models the effect of adversarial error injections, which is the injection metric [62]. This metric allows better design of nonconstant-dimension codes than the subspace metric.…”

Robust Wireless Network Coding – An Overview

“…The authors show that the codes proposed in [14] are a special case of the proposed family of codes. In [30], the error correction problem in both coherent and non-coherent NC is considered under an adversarial model. In particular, as far as non-coherent NC is concerned, the authors introduce a different metric with respect to [14], and prove that it yields a measure of code performance that is more precise, when a non-constant-dimension code is used, than [14].…”

“…The new metric is called injection metric. In [27], the authors introduce a Gilbert-Varshamov bound for the codes constructed in [30] according to the definition of injection metric. Moreover, the construction framework in [22] is exploited to obtain new non-constant-dimension codes, which are shown to contain a large number of codewords than comparable codes designed for the subspace metric.…”