This paper presents a novel approach to network coding for distribution of large files. Instead of the usual approach of splitting packets into disjoint classes (also known as generations) we propose ...the use of overlapping classes. The overlapping allows the decoder to alternate between Gaussian elimination and back substitution, simultaneously boosting the performance and reducing the decoding complexity. Our approach can be seen as a combination of fountain coding and network coding. Simulation results are presented that demonstrate the promise of our approach.
It is shown that the capacity of the channel modeled by (a discretized version of) the stochastic nonlinear Schr\"odinger (NLS) equation is upper-bounded by \(\log(1+\text{SNR})\) with ...\(\text{SNR}=\mathcal P_0/\sigma^2(z)\), where \(\mathcal P_0\) is the average input signal power and \(\sigma^2(z)\) is the total noise power up to distance \(z\). The result is a consequence of the fact that the deterministic NLS equation is a Hamiltonian energy-preserving dynamical system.
A probabilistic error model for random network coding is considered. An upper bound on capacity is obtained for any channel parameters, and asymptotic expressions are provided in the limit of long ...packet length and/or large field size. A simple and efficient coding scheme is provided that achieves capacity in both limiting cases. The scheme has zero error probability and a probability of failure that decreases exponentially both in the packet length and in the field size in bits.
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 \(F_{q^n}\).
The problem of designing new physical-layer network coding (PNC) schemes via lattice partitions is considered. Building on a recent work by Nazer and Gastpar, who demonstrated its asymptotic gain ...using information-theoretic tools, we take an algebraic approach to show its potential in non-asymptotic settings. We first relate Nazer-Gastpar's approach to the fundamental theorem of finitely generated modules over a principle ideal domain. Based on this connection, we generalize their code construction and simplify their encoding and decoding methods. This not only provides a transparent understanding of their approach, but more importantly, it opens up the opportunity to design efficient and practical PNC schemes. Finally, we apply our framework for PNC to a Gaussian relay network and demonstrate its advantage over conventional PNC schemes.
Application of constrained coding to 40-Gb/s dispersion-managed optical communication systems limited by intrachannel four-wave mixing is considered. When transmitted sequences obey the so-called (2, .../spl infin/) constraint, a 50% increase in data rate is demonstrated.
Staircase codes, a new class of forward-error-correction (FEC) codes suitable for high-speed optical communications, are introduced. An ITU-T G.709-compatible staircase code with rate R=239/255 is ...proposed, and FPGA-based simulation results are presented, exhibiting a net coding gain (NCG) of 9.41 dB at an output error rate of 1E-15, an improvement of 0.42 dB relative to the best code from the ITU-T G.975.1 recommendation. An error floor analysis technique is presented, and the proposed code is shown to have an error floor at 4.0E-21.
This paper considers the design of near capacity-achieving error correcting codes for a discrete multi-tone system in the presence of both additive white Gaussian noise and impulse noise. Impulse ...noise is one of the main channel impairments in the power-line channel. One way to combat impulse noise is to detect the presence of the impulses and to declare an erasure when an impulse occurs. In this paper, we propose a coding system based on irregular low-density parity-check (LDPC) codes and bit-interleaved coded modulation. We show that by carefully choosing the degree distribution of the irregular LDPC code, both the additive noise and the erasures can be handled in a single code. We show that the proposed system can perform close to the capacity of the channel and for the same redundancy is significantly more immune to the impulse noise than the existing methods based on an outer Reed-Solomon code.
The performance of low-density parity-check (LDPC) codes transmitted over a memoryless binary-input continuous output additive white Gaussian noise (AWGN) channel and decoded with quantized min-sum ...decoding is strongly influenced by the decoder's quantization scheme. This paper presents an efficient algorithm that determines the best uniform scalar quantizer for a particular code. To maximize performance, it is necessary to determine degree distributions that best match the characteristics of the quantized min-sum decoder. Toward this end, an iterative optimization framework that jointly optimizes the degree distributions and the quantizer is presented
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