Weight properties of network codes

In this paper, we first study the error correction and detection capability of codes for 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 characterization of the capability of a code for 1) error correction; 2) error detection; and 3) joint error correction and 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 characterized by a single minimum distance. By contrast, for a nonlinear code, two different minimum distances are needed for characterizing the capabilities of the code for error correction and for error detection. This leads to the surprising discovery that for a nonlinear code, the number of correctable errors can be more 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 in the same equivalence class of the Hamming weight and induce the same minimum distance as the Hamming weight. Copyright c © 0000 AEIT