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

Abstract

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 ǫ errors, µ erasures and δ deviations provided 2ǫwhere 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.

Orbit codes — A new concept in the area of network coding

“…In [1] a Reed-Solomon-like construction for constant dimension codes was introduced and Silva et al showed that lifting (i.e. concatenating an identity block in front of a matrix and taking the row space) maximum rank distance codes leads to exactly the same codes [4]. The cardinality of Reed-Solomon-like codes is q (n−k)(k−δ+1) for given ambient space dimension n, subspace dimension k and minimum distance 2δ.…”

“…Different approaches of constructing constant dimension codes have been investigated, e.g. in [1], [2], [3], [4], [5] and [6].…”

Erasure correction for nested receivers

“…Network coding for multicast packet erasure correction was considered in [4], [5]. The problem of multicast noncoherent error correction, where the network topology and/or network code are not known a priori, was studied in [6], [7], [8].…”

Secure network coding in the presence of eavesdroppers

“…The second attack model is called Byzantine attacks that attackers may modify the coded packets. To address this problem, Koetter [16] and Kschischang [17] proposed the rank metric error-correcting codes, then extended for the scenario in which the channel may supply partial information about erasures and deviations from the sent information flow [18]. At the same time, some network error correction codes have been proposed in [19][20][21].…”