Deux défis de la théorie du codage algébrique sont traités dans cette thèse. Le premier est le décodage efficace (dur et souple) de codes de Reed--Solomon généralisés sur les corps finis en métrique de Hamming. La motivation pour résoudre ce problème vieux de plus de 50 ans a été renouvelée par la découverte par Guruswami et Sudan à la fin du 20ème siècle d'un algorithme polynomial de décodage jusqu'au rayon Johnson basé sur l'interpolation. Les premières méthodes de décodage algébrique des codes de Reed--Solomon généralisés faisaient appel à une équation clé, c'est à dire, une description polynomiale du problème de décodage. La reformulation de l'approche à base d'interpolation en termes d'équations clés est un thème central de cette thèse. Cette contribution couvre plusieurs aspects des équations clés pour le décodage dur ainsi que pour la variante décodage souple de l'algorithme de Guruswami--Sudan pour les codes de Reed--Solomon généralisés. Pour toutes ces variantes un algorithme de décodage efficace est proposé. Le deuxième sujet de cette thèse est la formulation et le décodage jusqu'à certaines bornes inférieures sur leur distance minimale de codes en blocs linéaires cycliques. La caractéristique principale est l'intégration d'un code cyclique donné dans un code cyclique produit (généralisé). Nous donnons donc une description détaillée du code produit cyclique et des codes cycliques produits généralisés. Nous prouvons plusieurs bornes inférieures sur la distance minimale de codes cycliques linéaires qui permettent d'améliorer ou de généraliser des bornes connues. De plus, nous donnons des algorithmes de décodage d'erreurs/d'effacements [jusqu'à ces bornes] en temps quadratique.

This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed. A new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD. The optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution.

Since the frantic race towards the Shannon bound [1] commenced in the early 1950’s, linear block codes have become integral components of most digital communication systems. Both binary and non-binary linear block codes have proven themselves as formidable adversaries against the impediments presented by wireless communication channels. However, prior to the landmark 1974 paper [2] by Bahl et al. on the optimal Maximum a-Posteriori Probability (MAP) trellis decoding of linear block codes, practical linear block code decoding schemes were not only based on suboptimal hard decision algorithms, but also code-specific in most instances. In 1978 Wolf expedited the work of Bahl et al. by demonstrating the applicability of a block-wise Viterbi Algorithm (VA) to Bahl-Cocke-Jelinek-Raviv (BCJR) trellis structures as a generic optimal soft decision Maximum-Likelihood (ML) trellis decoding solution for linear block codes [3]. This study, largely motivated by code implementers’ ongoing search for generic linear block code decoding algorithms, builds on the foundations established by Bahl, Wolf and other contributing researchers by thoroughly evaluating the VA decoding of popular binary and non-binary linear block codes on realistic narrowband and wideband digital communication platforms in lifelike mobile environments. Ideally, generic linear block code decoding algorithms must not only be modest in terms of computational complexity, but they must also be channel aware. Such universal algorithms will undoubtedly be integrated into most channel coding subsystems that adapt to changing mobile channel conditions, such as the adaptive channel coding schemes of current Enhanced Data Rates for GSM Evolution (EDGE), 3rd Generation (3G) and Beyond 3G (B3G) systems, as well as future 4th Generation (4G) systems. In this study classic BCJR linear block code trellis construction is annotated and applied to contemporary binary and non-binary linear block codes. Since BCJR trellis structures are inherently sizable and intricate, rudimentary trellis complexity calculation and reduction algorithms are also presented and demonstrated. The block-wise VA for BCJR trellis structures, initially introduced by Wolf in [3], is revisited and improved to incorporate Channel State Information (CSI) during its ML decoding efforts. In order to accurately appraise the Bit-Error-Rate (BER) performances of VA decoded linear block codes in authentic wireless communication environments, Additive White Gaussian Noise (AWGN), flat fading and multi-user multipath fading simulation platforms were constructed. Included in this task was the development of baseband complex flat and multipath fading channel simulator models, capable of reproducing the physical attributes of realistic mobile fading channels. Furthermore, a complex Quadrature Phase Shift Keying (QPSK) system were employed as the narrowband communication link of choice for the AWGN and flat fading channel performance evaluation platforms. The versatile B3G multi-user multipath fading simulation platform, however, was constructed using a wideband RAKE receiver-based complex Direct Sequence Spread Spectrum Multiple Access (DS/SSMA) communication system that supports unfiltered and filtered Complex Spreading Sequences (CSS). This wideband platform is not only capable of analysing the influence of frequency selective fading on the BER performances of VA decoded linear block codes, but also the influence of the Multi-User Interference (MUI) created by other users active in the Code Division Multiple Access (CDMA) system. CSS families considered during this study include Zadoff-Chu (ZC) [4, 5], Quadriphase (QPH) [6], Double Sideband (DSB) Constant Envelope Linearly Interpolated Root-of- Unity (CE-LI-RU) filtered Generalised Chirp-like (GCL) [4, 7-9] and Analytical Bandlimited Complex (ABC) [7, 10] sequences. Numerous simulated BER performance curves, obtained using the AWGN, flat fading and multi-user multipath fading channel performance evaluation platforms, are presented in this study for various important binary and non-binary linear block code classes, all decoded using the VA. Binary linear block codes examined include Hamming and Bose-Chaudhuri-Hocquenghem (BCH) codes, whereas popular burst error correcting non-binary Reed-Solomon (RS) codes receive special attention. Furthermore, a simple cyclic binary linear block code is used to validate the viability of employing the reduced trellis structures produced by the proposed trellis complexity reduction algorithm. The simulated BER performance results shed light on the error correction capabilities of these VA decoded linear block codes when influenced by detrimental channel effects, including AWGN, Doppler spreading, diminished Line-of-Sight (LOS) signal strength, multipath propagation and MUI. It also investigates the impact of other pertinent communication system configuration alternatives, including channel interleaving, code puncturing, the quality of the CSI available during VA decoding, RAKE diversity combining approaches and CSS correlation characteristics. From these simulated results it can not only be gathered that the VA is an effective generic optimal soft input ML decoder for both binary and non-binary linear block codes, but also that the inclusion of CSI during VA metric calculations can fortify the BER performances of such codes beyond that attainable by classic ML decoding algorithms. / Dissertation (MEng(Electronic))--University of Pretoria, 2006. / Electrical, Electronic and Computer Engineering / unrestricted