The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be 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). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if $\mu$ erasures and $\delta$ deviations occur, then errors of rank $t$ can always be corrected provided that $2t \leq d - 1 + \mu + \delta$, where $d$ is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where $n$ packets of length $M$ over $F_q$ are transmitted, the complexity of the decoding algorithm is given by $O(dM)$ operations in an extension field $F_{q^n}$.
While random linear network coding [1]- [3] is an effective technique for information dissemination in communication networks, it is highly susceptible to errors. The insertion of even a single corrupt packet has the potential, when linearly combined with legitimate packets, to affect all packets gathered by an information receiver. The problem of error control in random network coding is therefore of great interest.
In this paper, we focus on end-to-end error control coding, where only the source and destination nodes apply error control techniques. Internal network nodes are assumed to be unaware of the presence of an outer code; they simply create outgoing packets as random linear combinations of incoming packets in the usual manner of random network coding. In addition, we assume that the source and destination nodes have no knowledge-or at least make no effort to exploit knowledge-of the topology of the network or of the particular network code used in the network. This is in contrast to the pioneering approaches [4]- [6], which have considered the design of a network code as part of the error control problem.
In the basic transmission model for end-to-end coding, the source node produces n packets, which are length-M vectors in a finite field F q , and the receiver gathers N packets. Additive packet errors may occur in any of the links. The channel equation is given by Y = AX + BZ, where X, Y and Z are matrices whose rows represent the transmitted, received and (possibly) corrupting packets, respectively, and A and B are the (unknown) corresponding transfer matrices induced by linear network coding.
There have been three previous quite different approaches to reliable communication under this model.
In [7], Zhang characterizes the error correction capability of a network code under a brute-force decoding algorithm. He shows that network codes with good error-correcting properties exist if the field size is sufficiently large. His approach can be applied to random network coding if an extended header is included in each packet in order to allow for the matrix B (as well as A) to be estimated at a sink node.
A drawback of this approach is that the extended header has size equal to the number of network edges, which may incur excessive overhead. In addition, no efficient decoding algorithm is provided for errors occurring according to an adversarial model. Jaggi et al. [8] propose a different approach specifically targeted to combat Byzantine adversaries.
They provide rate-optimal end-to-end codes that do not rely on the specific network code used and that can be decoded in polynomial time. However, their approach is based on probabilistic arguments that require both the field size and the packet length to be sufficiently large.
In contrast, Kötter and Kschischang [9] take a more combinatorial approach to the problem, which provides correction guarantees against adversarial errors and can be used with any given field and packet size. Their key observation is that, under the unknown linear transformation applied by random network coding, the only property of the matrix X that is preserved is its row space. Thus, information should be encoded in the choice of a subspace rather than a specific matrix. The receiver observes a subspace, given by the row space of Y , which may be different from the transmitted space when packet errors occur. A metric is proposed to account for the discrepancy between transmitted and received spaces, and a new coding theory based on this metric is developed. In particular, nearly-optimal Reed-Solomon-like codes are proposed that can be decoded in O(nM ) operations in an extension field F q n . Although the approach in [9] seems to be the appropriate abstraction of the error control problem in random network coding, one inherent difficulty is the absence of a natural group structure on the set of all subspaces of the ambient space F M q . As a consequence, many of the powerful concepts of classical coding theory such as group codes and linear codes do not naturally extend to codes consisting of subspaces.
In this paper, we explore the close relationship between subspace codes and codes for yet another distance measure: the rank metric. Codewords of a rank metric code are n × m matrices and the rank distance between two matrices is the rank of their difference. The rank metric was introduced in coding theory by Delsarte [10]. Codes for the rank metric were largely developed by Gabidulin [11] (see also [10], [12]). An important feature of the coding theory for the rank metric is that it supports many of the powerful concepts and techniques of classical coding theory, such as linear and cyclic codes and corresponding decoding algorithms [11]- [14].
One main contribution of this paper is to show that codes in the rank metric can be naturally “lifted” to subspace codes in such a way that the rank distance between two codewords is reflected in the subspace distance between th
This content is AI-processed based on open access ArXiv data.