Typical performance of gallager-type error-correcting codes

The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this method, the transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding.

TAP for parity check error correcting codes

We employ the methods presented in the previous chapter for decoding corrupted codewords, encoded using sparse parity check error correcting codes. We show the similarity between the equations derived from the TAP approach and those obtained from belief propagation, and examine their performance as practical decoding methods.

Statistical mechanics of low-density parity check error-correcting codes over Galois fields

A variation of low-density parity check (LDPC) error-correcting codes defined over Galois fields (GF(q)) is investigated using statistical physics. A code of this type is characterised by a sparse random parity check matrix composed of C non-zero elements per column. We examine the dependence of the code performance on the value of q, for finite and infinite C values, both in terms of the thermodynamical transition point and the practical decoding phase characterised by the existence of a unique (ferromagnetic) solution. We find different q-dependence in the cases of C = 2 and C ≥ 3; the analytical solutions are in agreement with simulation results, providing a quantitative measure to the improvement in performance obtained using non-binary alphabets.

Public key cryptography and error correcting codes as Ising models

We employ the methods of statistical physics to study the performance of Gallager type error-correcting codes. In this approach, the transmitted codeword comprises Boolean sums of the original message bits selected by two randomly-constructed sparse matrices. We show that a broad range of these codes potentially saturate Shannon's bound but are limited due to the decoding dynamics used. Other codes show sub-optimal performance but are not restricted by the decoding dynamics. We show how these codes may also be employed as a practical public-key cryptosystem and are of competitive performance to modern cyptographical methods.

Error-correcting codes on a Bethe-like lattice

We analyse Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as a minimization of a proper free-energy. We find a thermodynamical phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

Statistical physics of low density parity check error correcting codes

We study the performance of Low Density Parity Check (LDPC) error-correcting codes using the methods of statistical physics. LDPC codes are based on the generation of codewords using Boolean sums of the original message bits by employing two randomly-constructed sparse matrices. These codes can be mapped onto Ising spin models and studied using common methods of statistical physics. We examine various regular constructions and obtain insight into their theoretical and practical limitations. We also briefly report on results obtained for irregular code constructions, for codes with non-binary alphabet, and on how a finite system size effects the error probability.

Magnetization enumerator for real-valued symmetric channels in Gallager error-correcting codes

Using the magnetization enumerator method, we evaluate the practical and theoretical limitations of symmetric channels with real outputs. Results are presented for several regular Gallager code constructions.

Bayesian error bars for regression

Regression problems are concerned with predicting the values of one or more continuous quantities, given the values of a number of input variables. For virtually every application of regression, however, it is also important to have an indication of the uncertainty in the predictions. Such uncertainties are expressed in terms of the error bars, which specify the standard deviation of the distribution of predictions about the mean. Accurate estimate of error bars is of practical importance especially when safety and reliability is an issue. The Bayesian view of regression leads naturally to two contributions to the error bars. The first arises from the intrinsic noise on the target data, while the second comes from the uncertainty in the values of the model parameters which manifests itself in the finite width of the posterior distribution over the space of these parameters. The Hessian matrix which involves the second derivatives of the error function with respect to the weights is needed for implementing the Bayesian formalism in general and estimating the error bars in particular. A study of different methods for evaluating this matrix is given with special emphasis on the outer product approximation method. The contribution of the uncertainty in model parameters to the error bars is a finite data size effect, which becomes negligible as the number of data points in the training set increases. A study of this contribution is given in relation to the distribution of data in input space. It is shown that the addition of data points to the training set can only reduce the local magnitude of the error bars or leave it unchanged. Using the asymptotic limit of an infinite data set, it is shown that the error bars have an approximate relation to the density of data in input space.

Statistical physics of error-correcting codes

In this thesis we use statistical physics techniques to study the typical performance of four families of error-correcting codes based on very sparse linear transformations: Sourlas codes, Gallager codes, MacKay-Neal codes and Kanter-Saad codes. We map the decoding problem onto an Ising spin system with many-spins interactions. We then employ the replica method to calculate averages over the quenched disorder represented by the code constructions, the arbitrary messages and the random noise vectors. We find, as the noise level increases, a phase transition between successful decoding and failure phases. This phase transition coincides with upper bounds derived in the information theory literature in most of the cases. We connect the practical decoding algorithm known as probability propagation with the task of finding local minima of the related Bethe free-energy. We show that the practical decoding thresholds correspond to noise levels where suboptimal minima of the free-energy emerge. Simulations of practical decoding scenarios using probability propagation agree with theoretical predictions of the replica symmetric theory. The typical performance predicted by the thermodynamic phase transitions is shown to be attainable in computation times that grow exponentially with the system size. We use the insights obtained to design a method to calculate the performance and optimise parameters of the high performance codes proposed by Kanter and Saad.

Phase synchronization scheme for a practical phase sensitive amplifier of ASK-NRZ signals

We present a phase locking scheme that enables the demonstration of a practical dual pump degenerate phase sensitive amplifier for 10 Gbit/s non-return to zero amplitude shift keying signals. The scheme makes use of cascaded Mach Zehnder modulators for creating the pump frequencies as well as of injection locking for extracting the signal carrier and synchronizing the local lasers. An in depth optimization study has been performed, based on measured error rate performance, and the main degradation factors have been identified.

Developments towards practical free-space quantum cryptography

We describe a free space quantum cryptography system which is designed to allow continuous unattended key exchanges for periods of several days, and over ranges of a few kilometres. The system uses a four-laser faint-pulse transmission system running at a pulse rate of 10MHz to generate the required four alternative polarization states. The receiver module similarly automatically selects a measurement basis and performs polarization measurements with four avalanche photodiodes. The controlling software can implement the full key exchange including sifting, error correction, and privacy amplification required to generate a secure key.