The paradigm of network coding was introduced as a means to conserve bandwidth (or equivalently increase throughput) in information flow networks. Network coding makes use of the fact that unlike physical commodities, information can be replicated and coded together at the nodes of the network. As a result, routing can be strictly suboptimal in many classes of information flow networks compared to network coding. Network-error correction is the art of designing network codes such that the sinks of the network will be able to decode the required information in the presence of errors in the edges of the network, known as network-errors. The network coding problem on a network with given sink demands can be considered to have the following three major subproblems, which naturally also extend to the study of network-error correcting codes, as they can be viewed as a special class of network codes (a) Existence of a network code that satisfies the demands (b) Efficient construction of such a network code (c) Minimum alphabet size for the existence of such a network code. This thesis primarily considers linear network coding and error correction and in- vestigates solutions to these issues for certain classes of network coding and error correction problems in acyclic networks. Our contributions are broadly summarised as follows. (1) We propose the use of convolutional codes for multicast network-error correc- tion. Depending upon the number of network-errors required to be corrected in the network, convolutional codes are designed at the source of the multicast network so that these errors can be corrected at the sinks of the networks as long as they are separated by certain number of time instants (for which we give a bound). In con- trast to block codes for network-error correction which require large field sizes, using convolutional codes enables the field size of the network code to be small. We discuss the performance of such networks under the BSC edge error model. (2)Existing construction algorithms of block network-error correcting codes require a rather large field size, which grows with the size of the network and the number of sinks, and thereby can be prohibitive in large networks. In our work, we give an algorithm which, starting from a given network-error correcting code, can obtain an- other network code using a small field, with the same error correcting capability as the original code. The major step in our algorithm is to find a least degree irreducible poly- nomial which is coprime to another large degree polynomial. We utilize the algebraic properties of finite fields to implement this step so that it becomes much faster than the brute-force method. A recently proposed algorithm for network coding using small fields can be seen as a special case of our algorithm for the case of no network-errors. (3)Matroids are discrete mathematical objects which generalize the notion of linear independence of sets of vectors. It has been observed recently that matroids and network coding share a deep connection, and several important results of network coding has been obtained using these connections from matroid theory. In our work, we establish that matroids with certain special properties correspond to networks with error detecting and correcting properties. We call such networks as matroidal error detecting (or equivalently, correcting) networks. We show that networks have scalar linear network-error detecting (or correcting) codes if and only if there are associated with representable matroids with some special properties. We also use these ideas to construct matroidal error correcting networks along with their associated matroids. In the case of representable matroids, these algorithms give rise to scalar linear network- error correcting codes on such networks. Finally we also show that linear network coding is not sufficient for the general network-error detection (correction) problem with arbitrary demands. (4)Problems related to network coding for acyclic, instantaneous networks have been extensively dealt with in the past. In contrast, not much attention has been paid to networks with delays. In our work, we elaborate on the existence, construction and minimum field size issues of network codes for networks with integer delays. We show that the delays associated with the edges of the network cannot be ignored, and in fact turn out to be advantageous, disadvantageous or immaterial, depending on the topology of the network and the network coding problem considered. In the process, we also show multicast network codes which involve only delaying the symbols arriving at the nodes of the networks and coding the delayed symbols over a binary field, thereby making coding operations at the nodes less complex. (5) In the usual network coding framework, for a given set of network demands over an arbitrary acyclic network with integer delays assumed for the links, the out- put symbols at the sink nodes, at any given time instant, is a Fq-linear combination of the input symbols generated at different time instants where Fq denotes the field over which the network operates. Therefore the sinks have to use sufficient memory elements in order to decode simultaneously for the entire stream of demanded infor- mation symbols. We propose a scheme using an ν-point finite-field discrete fourier transform (DFT) which converts the output symbols at the sink nodes at any given time instant, into a Fq-linear combination of the input symbols generated during the same time instant without making use of memory at the intermediate nodes. We call this as transforming the acyclic network with delay into ν-instantaneous networks (ν is sufficiently large). We show that under certain conditions, there exists a network code satisfying sink demands in the usual (non-transform) approach if and only if there exists a network code satisfying sink demands in the transform approach.

A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.

Thesis (MScEng)-- Stellenbosch University, 2013. / ENGLISH ABSTRACT: All process measurements contain some element of error. Typically, a distinction is made between random errors, with zero expected value, and gross errors with non-zero magnitude. Data Reconciliation (DR) and Gross Error Detection (GED) comprise a collection of techniques designed to attenuate measurement errors in process data in order to reduce the effect of the errors on subsequent use of the data. DR proceeds by finding the optimum adjustments so that reconciled measurement data satisfy imposed process constraints, such as material and energy balances. The DR solution is optimal under the assumed statistical random error model, typically Gaussian with zero mean and known covariance. The presence of outliers and gross errors in the measurements or imposed process constraints invalidates the assumptions underlying DR, so that the DR solution may become biased. GED is required to detect, identify and remove or otherwise compensate for the gross errors. Typically GED relies on formal hypothesis testing of constraint residuals or measurement adjustment-based statistics derived from the assumed random error statistical model. Classification methodologies are methods by which observations are classified as belonging to one of several possible groups. For the GED problem, artificial neural networks (ANN’s) have been applied historically to resolve the classification of a data set as either containing or not containing a gross error. The hypothesis investigated in this thesis is that classification methodologies, specifically classification trees (CT) and linear or quadratic classification functions (LCF, QCF), may provide an alternative to the classical GED techniques. This hypothesis is tested via the modelling of a simple steady-state process unit with associated simulated process measurements. DR is performed on the simulated process measurements in order to satisfy one linear and two nonlinear material conservation constraints. Selected features from the DR procedure and process constraints are incorporated into two separate input vectors for classifier construction. The performance of the classification methodologies developed on each input vector is compared with the classical measurement test in order to address the posed hypothesis. General trends in the results are as follows: - The power to detect and/or identify a gross error is a strong function of the gross error magnitude as well as location for all the classification methodologies as well as the measurement test. - For some locations there exist large differences between the power to detect a gross error and the power to identify it correctly. This is consistent over all the classifiers and their associated measurement tests, and indicates significant smearing of gross errors. - In general, the classification methodologies have higher power for equivalent type I error than the measurement test. - The measurement test is superior for small magnitude gross errors, and for specific locations, depending on which classification methodology it is compared with. There is significant scope to extend the work to more complex processes and constraints, including dynamic processes with multiple gross errors in the system. Further investigation into the optimal selection of input vector elements for the classification methodologies is also required. / AFRIKAANSE OPSOMMING: Alle prosesmetings bevat ʼn sekere mate van metingsfoute. Die fout-element van ʼn prosesmeting word dikwels uitgedruk as bestaande uit ʼn ewekansige fout met nul verwagte waarde, asook ʼn nie-ewekansige fout met ʼn beduidende grootte. Data Rekonsiliasie (DR) en Fout Opsporing (FO) is ʼn versameling van tegnieke met die doelwit om die effek van sulke foute in prosesdata op die daaropvolgende aanwending van die data te verminder. DR word uitgevoer deur die optimale veranderinge aan die oorspronklike prosesmetings aan te bring sodat die aangepaste metings sekere prosesmodelle gehoorsaam, tipies massa- en energie-balanse. Die DR-oplossing is optimaal, mits die statistiese aannames rakende die ewekansige fout-element in die prosesdata geldig is. Dit word tipies aanvaar dat die fout-element normaal verdeel is, met nul verwagte waarde, en ʼn gegewe kovariansie matriks. Wanneer nie-ewekansige foute in die data teenwoordig is, kan die resultate van DR sydig wees. FO is daarom nodig om nie-ewekansige foute te vind (Deteksie) en te identifiseer (Identifikasie). FO maak gewoonlik staat op die statistiese eienskappe van die meting aanpassings wat gemaak word deur die DR prosedure, of die afwykingsverskil van die model vergelykings, om formele hipoteses rakende die teenwoordigheid van nie-ewekansige foute te toets. Klassifikasie tegnieke word gebruik om die klasverwantskap van observasies te bepaal. Rakende die FO probleem, is sintetiese neurale netwerke (SNN) histories aangewend om die Deteksie en Identifikasie probleme op te los. Die hipotese van hierdie tesis is dat klassifikasie tegnieke, spesifiek klassifikasiebome (CT) en lineêre asook kwadratiese klassifikasie funksies (LCF en QCF), suksesvol aangewend kan word om die FO probleem op te los. Die hipotese word ondersoek deur middel van ʼn simulasie rondom ʼn eenvoudige gestadigde toestand proses-eenheid wat aan een lineêre en twee nie-lineêre vergelykings onderhewig is. Kunsmatige prosesmetings word geskep met behulp van lukrake syfers sodat die foutkomponent van elke prosesmeting bekend is. DR word toegepas op die kunsmatige data, en die DR resultate word gebruik om twee verskillende insetvektore vir die klassifikasie tegnieke te skep. Die prestasie van die klassifikasie metodes word vergelyk met die metingstoets van klassieke FO ten einde die gestelde hipotese te beantwoord. Die onderliggende tendense in die resultate is soos volg: - Die vermoë om ‘n nie-ewekansige fout op te spoor en te identifiseer is sterk afhanklik van die grootte asook die ligging van die fout vir al die klassifikasie tegnieke sowel as die metingstoets. - Vir sekere liggings van die nie-ewekansige fout is daar ‘n groot verskil tussen die vermoë om die fout op te spoor, en die vermoë om die fout te identifiseer, wat dui op smering van die fout. Al die klassifikasie tegnieke asook die metingstoets baar hierdie eienskap. - Oor die algemeen toon die klassifikasie metodes groter sukses as die metingstoets. - Die metingstoets is meer suksesvol vir relatief klein nie-ewekansige foute, asook vir sekere liggings van die nie-ewekansige fout, afhangende van die klassifikasie tegniek ter sprake. Daar is verskeie maniere om die bestek van hierdie ondersoek uit te brei. Meer komplekse, niegestadigde prosesse met sterk nie-lineêre prosesmodelle en meervuldige nie-ewekansige foute kan ondersoek word. Die moontlikheid bestaan ook om die prestasie van klassifikasie metodes te verbeter deur die gepaste keuse van insetvektor elemente.

<p>This thesis shows how certain classes of binary constant weight codes can be represented geometrically using linear structures in Euclidean space. The geometric treatment is concerned mostly with codes with minimum distance 2(w - 1), that is, where any two codewords coincide in at most one entry; an algebraic generalization of parts of the theory also applies to some codes without this property. The presented theorems lead to a total of 18 improvements of the table of lower bounds on A(n,d,w) maintained by E. M. Rains and N. J. A. Sloane. Additional improvements have been made by finding new lexicographic codes.</p>

Iris recognition constitutes one of the most powerful method for the iden- tification and authentication of people today. This thesis aims to describe the algorithms used in a sophisticated and mathematically correct way, while re- maining comprehensible. The description of these algorithms is not the only objective of this thesis; the reason they were chosen and potential improvements or substitutions are also discussed. The background of iris recognition, its use in cryptosystems, and the application of error-correcting codes are investigated as well.

Iris recognition constitutes one of the most powerful method for the iden- tification and authentication of people today. This thesis aims to describe the algorithms used by a mathematical apparatus. The description of these algo- rithms is not the only objective of this thesis; the reason they were chosen and potential improvements or substitutions are also discussed. The background of iris recognition, its use in cryptosystems, and the application of error-correcting codes are investigated as well. The second version of the thesis eliminates errata and a quantum of inaccu- racies discovered in the first version, especially in the ROI Definition, the Hough Transform and the Feature Extraction sections. Besides that, it also contains se- veral new propositions. Last, but not least, it shows a potential implementation of the algorithms described by appending pseudocodes to the relevant sections. 1

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Masters of Science in Engineering (Electrical). Johannesburg, 2016 / Power-line communications can be used for the transfer of data across electrical net- works in applications such as automatic meter reading in smart grid technology. As the power-line channel is harsh and plagued with non-Gaussian noise, robust forward error correction schemes are required. This research is a comparative study where a Luby transform code is concatenated with power-line communication systems provided by an up-to-date standard published by electricit e R eseau Distribution France named G3 PLC. Both decoding using Gaussian elimination and belief propagation are imple- mented to investigate and characterise their behaviour through computer simulations in MATLAB. Results show that a bit error rate performance improvement is achiev- able under non worst-case channel conditions using a Gaussian elimination decoder. An adaptive system is thus recommended which decodes using Gaussian elimination and which has the appropriate data rate. The added complexity can be well tolerated especially on the receiver side in automatic meter reading systems due to the network structure being built around a centralised agent which possesses more resources. / MT2017

A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy in the School of Electrical and Information Engineering, 2016 / Reliable and e cient data transmission have been the subject of current research, most especially in realistic channels such as the Rayleigh fading channels. The focus of every new technique is to improve the transmission reliability and to increase the transmission capacity of the communication links for more information to be transmitted. Modulation schemes such as M-ary Quadrature Amplitude Modulation (M-QAM) and Orthogonal Frequency Division Multiplexing (OFDM) were developed to increase the transmission capacity of communication links without additional bandwidth expansion, and to reduce the design complexity of communication systems. On the contrary, due to the varying nature of communication channels, the message transmission reliability is subjected to a couple of factors. These factors include the channel estimation techniques and Forward Error Correction schemes (FEC) used in improving the message reliability. Innumerable channel estimation techniques have been proposed independently, and in combination with di erent FEC schemes in order to improve the message reliability. The emphasis have been to improve the channel estimation performance, bandwidth and power consumption, and the implementation time complexity of the estimation techniques. Of particular interest, FEC schemes such as Reed-Solomon (RS) codes, Turbo codes, Low Density Parity Check (LDPC) codes, Hamming codes, and Permutation codes, are proposed to improve the message transmission reliability of communication links. Turbo and LDPC codes have been used extensively to combat the varying nature of communication channels, most especially in joint iterative channel estimation and decoding receiver structures. In this thesis, attention is focused on using RS codes to improve the message reliability of a communication link because RS codes have good capability of correcting random and burst errors, and are useful in di erent wireless applications. This study concentrates on symbol level soft decision decoding of RS codes. In this regards, a novel symbol level iterative soft decision decoder for RS codes based on parity-check equations is developed. This Parity-check matrix Transformation Algorithm (PTA) is based on the soft reliability information derived from the channel output in order to perform syndrome checks in an iterative process. Performance analysis verify that this developed PTA outperforms the conventional RS hard decision decoding algorithms and the symbol level Koetter and Vardy (KV ) RS soft decision decoding algorithm. In addition, this thesis develops an improved Distance Metric (DM) method of deriving reliability information over Rayleigh fading channels for combined demodulation with symbol level RS soft decision decoding algorithms. The newly proposed DM method incorporates the channel state information in deriving the soft reliability information over Rayleigh fading channels. Analysis verify that this developed metric enhances the performance of symbol level RS soft decision decoders in comparison with the conventional method. Although, in this thesis, the performance of the developed DM method of deriving soft reliability information over Rayleigh fading channels is only veri ed for symbol level RS soft decision decoders, it is applicable to any symbol level soft decision decoding FEC scheme. Besides, the performance of the all FEC decoding schemes plummet as a result of the Rayleigh fading channels. This engender the development of joint iterative channel estimation and decoding receiver structures in order to improve the message reliability, most especially with Turbo and LDPC codes as the FEC schemes. As such, this thesis develops the rst joint iterative channel estimation and Reed- Solomon decoding receiver structure. Essentially, the joint iterative channel estimation and RS decoding receiver is developed based on the existing symbol level soft decision KV algorithm. Consequently, the joint iterative channel estimation and RS decoding receiver is extended to the developed RS parity-check matrix transformation algorithm. The PTA provides design ease and exibility, and lesser computational time complexity in an iterative receiver structure in comparison with the KV algorithm. Generally, the ndings of this thesis are relevant in improving the message transmission reliability of a communication link with RS codes. For instance, it is pertinent to numerous data transmission technologies such as Digital Audio Broadcasting (DAB), Digital Video Broadcasting (DVB), Digital Subscriber Line (DSL), WiMAX, and long distance satellite communications. Equally, the developed, less computationally intensive, and performance e cient symbol level decoding algorithm for RS codes can be use in consumer technologies like compact disc and digital versatile disc. / GS2016

A thesis submitted in ful lment of the requirements for the degree of Master of Science in the Centre of Excellence in Telecommunications and Software School of Electrical and Information Engineering, March 2016 / In order to compare low rate error correcting codes to MIMO schemes with transmit diversity, two systems with the same throughput are compared. A VBLAST MIMO system with (15; 5) Reed-Solomon coding is compared to an Alamouti MIMO system with (15; 10) Reed-Solomon coding. The latter is found to perform signi cantly better, indicating that transmit diversity is a more e ective technique for minimising errors than reducing the code rate. The Guruswami-Sudan/Koetter-Vardy soft decision decoding algorithm was implemented to allow decoding beyond the conventional error correcting bound of RS codes and VBLAST was adapted to provide reliability information. Analysis is also performed to nd the optimal code rate when using various MIMO systems. / MT2016

Este trabalho, elaborado por meio de pesquisa bibliográfica, apresenta um apanhado sobre os dígitos verificadores presentes no Cadastro de Pessoas Físicas (CPF), no código de barras, e no sistema ISBN; faz uma introdução sobre a métrica de Hamming e os códigos corretores de erros; cita a classe de códigos mais utilizada, que são os códigos lineares, e deixa a sugestão de uma proposta pedagógica para professores de matemática aplicarem no Ensino Fundamental, podendo ser ajustada também para o Ensino Médio. No apêndice A, são propostos alguns exercícios que podem ser trabalhados com os alunos em sala de aula. / This work, based on the attached references, presents an overview of the check digits that appear in the Brazilian document CPF, in the bar code and the ISBN system. Moreover, it makes an introduction to the Hamming metric and error-correcting codes. In particular, some considerations about linear codes are done and it makes a suggestion of a pedagogical approach to apply it in middle school and can also be adjusted to high school. In the Appendix A are proposed some exercises to students.