A Rank-Metric Approach to Error Control in Random Network Coding

It is shown that the error control problem in random network coding can be reformulated as a generalized decoding problem for rank-metric codes. This result allows many of the tools developed for rank-metric codes to be applied to random network coding. In the generalized decoding problem induced by random network coding, the channel may supply partial information about the error in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can fully exploit the correction capability of the code; namely, it can correct any pattern of $\epsilon$ errors, $\mu$ erasures and $\delta$ deviations provided $2\epsilon + \mu + \delta \leq d-1$, where $d$ is the minimum rank distance of the code. Our approach is based on the coding theory for subspaces introduced by Koetter and Kschischang and can be seen as a practical way to construct codes in that context.