The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes - nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.

During the race towards the4th generation (4G) cellular-based digital communication systems, a growth in the demand for high capacity, multi-media capable, improved Quality-of-Service (QoS) mobile communication systems have caused the developing mobile communications world to turn towards betterMultiple Access (MA) techniques, like Code Division Multiple Access (CDMA) [5]. The demand for higher throughput and better QoS in future 4G systems have also given rise to a scheme that is becoming ever more popular for use in these so-called ‘bandwidth-on-demand’ systems. This scheme is known as adaptive channel coding, and gives a system the ability to firstly sense changes in conditions, and secondly, to adapt to these changes, exploiting the fact that under good channel conditions, a very simple or even no channel coding scheme can be used for Forward Error Correction(FEC). This will ultimately result in better system throughput utilization. One such scheme, known as incremental redundancy, is already implemented in the Enhanced Data Rates for GSM Evolution (EDGE) standard. This study presents an extensive simulation study of a Multi-User (MU), adaptive channel coded Direct Sequence Spread Spectrum Multiple Access (DS/SSMA) communication system. This study firstly presents and utilizes a complex Base Band(BB) DS/SSMA transmitter model, aimed at user data diversity [6] in order to realize the MU input data to the system. This transmitter employs sophisticated double-sideband (DSB)Constant-Envelope Linearly Interpolated Root-of-Unity (CE-LI-RU) filtered General Chirp-Like (GCL) sequences [34, 37, 38] to band limit and spread user data. It then utilizes a fully user-definable, complex Multipath Fading Channel Simulator(MFCS), first presented by Staphorst [3], which is capable of reproducing all of the physical attributes of realistic mobile fading channels. Next, this study presents a matching DS/SSMA receiver structure that aims to optimally recover user data from the channel, ensuring the achievement of data diversity. In order to provide the basic channel coding functionality needed by the system of this study, three simple, but well-known channel coding schemes are investigated and employed. These are: binary Hamming (7,4,3) block code, (15,7,5) binary Bose-Chadhuri-Hocquenghem (BCH) block code and a rate 1/3 <i.Non-Systematic (NS) binary convolutional code [6]. The first step towards the realization of any adaptive channel coded system is the ability to measure channel conditions as fast as possible, without the loss of accuracy or inclusion of known data. In 1965, Gooding presented a paper in which he described a technique that measures communication conditions at the receiving end of a system through a device called a Performance Monitoring Unit (PMU) [12, 13]. This device accelerates the system’sBit Error Rate (BER) to a so-called Pseudo Error Rate(PER) through a process known as threshold modification. It then uses a simple PER extrapolation algorithm to estimate the system’s true BER with moderate accuracy and without the need for known data. This study extends the work of Gooding by applying his technique to the DS/SSMA system that utilizes a generic Soft-Output Viterbi Algorithm(SOVA) decoder [39] structure for the trellis decoding of the binary linear block codes [3, 41-50], as well as binary convolutional codes mentioned, over realistic MU frequency selective channel conditions. This application will grant the system the ability to sense changes in communication conditions through real-time BER measurement and, ultimately, to adapt to these changes by switching to different channel codes. Because no previous literature exists on this application, this work is considered novel. Extensive simulation results also investigate the linearity of the PER vs. modified threshold relationship for uncoded, as well as all coded cases. These simulations are all done for single, as well as multiple user systems. This study also provides extensive simulation results that investigate the calculation accuracy and speed advantages that Gooding’s technique possesses over that of the classic Monte-Carlo technique for BER estimation. These simulations also consider uncoded and coded cases, as well as single and multiple users. Finally, this study investigates the experimental real-time performance of the fully functional MU, adaptive coded, DS/SSMA communication system over varying channel conditions. During this part of the study, the channel conditions are varied over time, and the system’s adaptation (channel code switching) performance is observed through a real-time observation of the system’s estimated BER. This study also extends into cases with multiple system users. Since the adaptive coded system of this study does not require known data sequences (training sequences), inclusion of Gooding’s technique for real-time BER estimation through threshold modification and PER extrapolation in future 4G adaptive systems will enable better Quality-of-Service (QoS) management without sacrificing throughput. Furthermore, this study proves that when Gooding’s technique is applied to a coded system with a soft-output, it can be an effective technique for QoS monitoring, and should be considered in 4G systems of the future. / Dissertation (MEng (Computer Engineering))--University of Pretoria, 2007. / Electrical, Electronic and Computer Engineering / MEng / unrestricted

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