Weight properties of network codes

In this paper, we first study the error correction and detection capability of codesfor a general transmission system inspired by network error correction. For a given weight measure on the error vectors, we define a corresponding minimum weight decoder. Then we obtain a complete characterisation of the capability of a code for (1) error correction; (2) error detection and (3) joint error correctionand detection. Our results show that if the weight measure on the error vectors is the Hamming weight, the capability of a linear code is fully characterised by a single minimum distance. By contrast, for a nonlinear code, two different minimum distances are needed for characterising the capabilities of the code for error correction and for errordetection. This leads to the surprising discovery that for a nonlinear code, the numberof correctable errors can bemore than half of the number of detectable errors. We also present a framework that captures joint error correction and detection. We further define equivalence classes of weight measures with respect to a channel. Specifically, for any given code, the minimum weight decoders for two different weight measures are equivalent if the two weight measures belong to the same equivalence class. In the special case of linear network coding, we study three weight measures, and show that they are inthe same equivalence class of the Hamming weight and induce the same minimum distance as the Hamming weight. Copyright © 2008 John Wiley & Sons, Ltd.