Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space $V$ and the collection by the receiver of a basis for a vector space $U$. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space $V \cap U$ is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification

Space-time coding techniques with bit-interleaved coded modulations for MIMO block-fading channels

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.Comment: Submitted to IEEE Trans. on Information Theory, Submission: January 2006 - First review: June 200

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

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 $F_{q^n}$.Comment: Minor corrections; 42 pages, to be published at the IEEE Transactions on Information Theor