Characterization of error correction and detection in a general transmission system

In this paper, we study the error correction and detection capabilities of block 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 capabilities of a block code for error correction and error detection. Our results imply that for a linear network code with the Hamming weight being the weight measure on the error vectors, the capability of the code is fully characterized by a single minimum distance. By contrast, for a nonlinear network 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 network code, the number of correctable errors can be more than half of the number of detectable errors. We further define equivalence classes of weight measures with respect to a channel. Specifically, for any given code, the minimum distance decoders for two different weight measures are equivalent if the two weight measures belong to the same equivalence class.