On the Decoder Error Probability of Rank Metric Codes and Constant-Dimension Codes

Rank metric codes can either be used as such for error correction in data storage equipments, or be lifted into constant-dimension codes (CDCs) and thus be used for error correction in random network coding. This paper investigates the decoder error probability (DEP) of rank metric codes and CDCs. We first study the DEP of rank metric codes using a bounded rank distance decoder. We derive asymptotically tight upper bounds on the DEP of rank metric codes used over an equal row or an equal column space channel, and we determine the exact DEP of maximum rank distance codes used over an equal row space channel. We then investigate the DEP of CDCs using a bounded subspace distance or a bounded modified subspace distance decoder over a symmetric operator channel. We first determine some fundamental properties of the subspace and the modified subspace metrics. Using these properties, we derive asymptotically tight upper bounds on the DEP of any CDC obtained by lifting a rank metric code using either a bounded subspace distance or a bounded modified subspace distance decoder.