On the decoder error probability of bounded rank distance decoders for rank metric codes

Abstract

In this paper we investigate the decoder error probability (DEP) of bounded rank distance decoders for rank metric codes over two types of channels motivated by network coding. The first channel is a rank symmetric channel where additive errors with the same rank are equiprobable, and for the second and more general channel, errors with the same row or column space are equiprobable. For arbitrary rank metric codes, we first derive analytical expressions of as well as upper bounds on DEPs of bounded distance decoders over the rank symmetric channel, and then establish upper bounds on DEP of bounded distance decoders over the equal row (or column) space channel. Our results show that DEP of bounded distance decoders for any rank metric code with error correction capability t decreases exponentially with t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . For maximum rank distance (MRD) codes, we determine the exact DEP of bounded distance decoders over equal row (or column) space channels as long as MRD codes or their transpose exist, and show that MRD codes have the highest DEP up to a scalar. These results provide insights on the error performance of rank metric codes used for error correction in random linear network coding.