Error-correction on non-standard communication channels

Many communication systems are poorly modelled by the standard channels assumed in the information theory literature, such as the binary symmetric channel or the additive white Gaussian noise channel. Real systems suffer from additional problems including time-varying noise, cross-talk, synchronization errors and latency constraints. In this thesis, low-density parity-check codes and codes related to them are applied to non-standard channels. First, we look at time-varying noise modelled by a Markov channel. A low-density parity-check code decoder is modified to give an improvement of over 1dB. Secondly, novel codes based on low-density parity-check codes are introduced which produce transmissions with Pr(bit = 1) ≠ Pr(bit = 0). These non-linear codes are shown to be good candidates for multi-user channels with crosstalk, such as optical channels. Thirdly, a channel with synchronization errors is modelled by random uncorrelated insertion or deletion events at unknown positions. Marker codes formed from low-density parity-check codewords with regular markers inserted within them are studied. It is shown that a marker code with iterative decoding has performance close to the bounds on the channel capacity, significantly outperforming other known codes. Finally, coding for a system with latency constraints is studied. For example, if a telemetry system involves a slow channel some error correction is often needed quickly whilst the code should be able to correct remaining errors later. A new code is formed from the intersection of a convolutional code with a high rate low-density parity-check code. The convolutional code has good early decoding performance and the high rate low-density parity-check code efficiently cleans up remaining errors after receiving the entire block. Simulations of the block code show a gain of 1.5dB over a standard NASA code

Convolutional Codes in Rank Metric with Application to Random Network Coding

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding.Comment: presented in part at Netcod 2012, submitted to IEEE Transactions on Information Theor

Competitive minimax universal decoding for several ensembles of random codes

Universally achievable error exponents pertaining to certain families of channels (most notably, discrete memoryless channels (DMC's)), and various ensembles of random codes, are studied by combining the competitive minimax approach, proposed by Feder and Merhav, with Chernoff bound and Gallager's techniques for the analysis of error exponents. In particular, we derive a single--letter expression for the largest, universally achievable fraction $\xi$ of the optimum error exponent pertaining to the optimum ML decoding. Moreover, a simpler single--letter expression for a lower bound to $\xi$ is presented. To demonstrate the tightness of this lower bound, we use it to show that $\xi=1$, for the binary symmetric channel (BSC), when the random coding distribution is uniform over: (i) all codes (of a given rate), and (ii) all linear codes, in agreement with well--known results. We also show that $\xi=1$ for the uniform ensemble of systematic linear codes, and for that of time--varying convolutional codes in the bit-error--rate sense. For the latter case, we also show how the corresponding universal decoder can be efficiently implemented using a slightly modified version of the Viterbi algorithm which em employs two trellises.Comment: 41 pages; submitted to IEEE Transactions on Information Theor

Localized Dimension Growth in Random Network Coding: A Convolutional Approach

We propose an efficient Adaptive Random Convolutional Network Coding (ARCNC) algorithm to address the issue of field size in random network coding. ARCNC operates as a convolutional code, with the coefficients of local encoding kernels chosen randomly over a small finite field. The lengths of local encoding kernels increase with time until the global encoding kernel matrices at related sink nodes all have full rank. Instead of estimating the necessary field size a priori, ARCNC operates in a small finite field. It adapts to unknown network topologies without prior knowledge, by locally incrementing the dimensionality of the convolutional code. Because convolutional codes of different constraint lengths can coexist in different portions of the network, reductions in decoding delay and memory overheads can be achieved with ARCNC. We show through analysis that this method performs no worse than random linear network codes in general networks, and can provide significant gains in terms of average decoding delay in combination networks.Comment: 7 pages, 1 figure, submitted to IEEE ISIT 201