Le sujet porte sur une classe de codes correcteurs d erreurs dits codes cycliques tordus, et ses applications a l'Informatique quantique et aux codes quasi-cycliques. Les codes cycliques classiques ont une structure d'idéaux dans un anneau de polynômes. Ulmer a introduit en 2008 une généralisation aux anneaux dits de polynômes tordus, une classe d'anneaux non commutatifs introduits par Ore en 1933. Dans cette thèse on explore le cas du corps a quatre éléments et de l'anneau produit de deux copies du corps a deux éléments. / The topic of the thesis is the study of skew cyclic codes, with application to Quantum Computing and quasi-cyclic codes. Classical cyclic codes have a natural structure of ideals in a polynomial ring. This was generalized by Ulmer in 2008 to skew polynomial rings, a class of non commutative rings introduced by Ore in 1933. The latter codes are not classically cyclic if the alphabet ring admits a non trivial automorphism. In this work is explored the cases of the finite field of order four and of a product ring of two copies of the finite field of order two.

M.Ing.(Electrical and Electronic Engineering) / In the ideal communication system no noise is present and no errors will be made. However, in practice, communication is over noisy channels which cause errors in the information. There is thus a necessity for the control of these errors. Furthermore, several channels impose runlength or disparity constraints on the bit stream. Until recently, the error control on these channels was applied separately to imposing the input restrictions with constrained codes. Such a system leads to poor performance under certain conditions. and is more complex and expensive to apply than systems where the error control is an integral part of the constrained code or decoder. In this study, we firstly investigate the error multiplication phenomena of constrained codes. An algorithm is presented that minimizes the error propagation probabilities of memoryless decoders according to two criteria. Another algorithm is presented along with the first to calculate the resulting bit error probabilities. The second approach to the error control of constrained codes is the construction of combined error-correcting constrained finite-state machine codes. We investigate the known construction techniques and construct several new codes using extensions of the known techniques. These codes complement or improve on the known error-correcting constrained codes with regards to either complexity, rate or error-correcting capability. Furthermore, these codes have good error behaviour and favourable power spectral densities.

D. Ing. / The digital transmission of information necessitates the compensation for disturbances introduced by the channel. The compensation method usually used in digital communications is error correcting coding. The errors usually encountered are additive in nature, i.e. errors where only symbol values are changed. Understandably, the field of additive error correcting codes has become a mature research field. Remarkable progress has been made during the past 50 years, to such an extent that near Shannon capacity can be reached using suitable coding techniques. Sometimes the channel disturbances may result in the loss and/or gain of symbols and a subsequent loss of word or frame synchronisation. Unless some precautions were made, a synchronisation error may propagate and corrupt large blocks of data. Typical precautions taken against synchronisation errors are: out-of-band clock signals distributed to the transmission equipment in a network; stringent requirements on clock stability and jitter; limits on the number of repeaters and regeneration to curb jitter and delays; line coding to facilitate better clock extraction; and - use of framing methods on the coding level. Most transmission systems in use today will stop data transmission until reliable synchronisation is restored. El multiplexing systems are still the predominantly used technology in fixed telephone line operators and GSM operators, and recovering from a loss of synchronisation (the FAS alarm) typically lasts approximately 10 seconds. Considering that the transmission speed is 2048 KB/s, a large quantity of data is lost in during this process. The purpose of this study is therefore to broaden the understanding of insertion/deletion correcting binary codes. This will be achieved by presenting new properties and coding techniques for multiple insertion/deletion correcting codes. Mostly binary codes will be considered, but in some instances, the results may also hold for non-binary codes. As a secondary purpose, we hope to generate interest in this field of study and enable other researchers to continue to deeper explore the mechanisms of insertion and/or deletion correcting codes.

Dans cette thèse, nous avons envisagé l’utilisation des codes en métrique rang pour des applications de communication sans fil en général, et les réseaux de capteurs en particulier. Après avoir introduit les codes en métrique rang, ces codes, qui ont été proposés dans le contexte de la cryptographie, sont adaptés par la suite pour la correction d’erreurs. Pour cela, une étude est faite sur le comportement de ces familles de codes dans un scénario de transmission sans fil en utilisant le codage réseau. Dans ce contexte, trois types d’erreurs sont considérés : le bruit de fond, les erreurs injectés dans le réseau par un utilisateur malveillant et les effacements qui peuvent être dus aux pannes des nœuds. L’analyse qui a été faite sur la famille des codes Low Rank Parity Check (LRPC) a montré que ces derniers sont plus adaptés aux réseaux de capteurs sans fil par rapport aux codes Gabidulin utilisés dans la littérature. Cette analyse a été généralisée dans le contexte multi-sources et a montré que les codes LRPC sont plus efficaces dans ce contexte. Ces contributions apportent un nouveau souffle à l’utilisation des codes en métrique rang et offrent des perspectives de poursuite intéressantes. / In this thesis, we have considered the rank metric codes for wireless sensor networks. Firstly, we have introduced the rank metric codes. Then, we adapted these codes, which were originally dedicated to cryptography applications, for error correction. To this end, we have studied the behavior of the family of rank metric codes in a wireless communication scenario using network coding. In this context, three types of errors are considered, background noise, errors injected into the network by a malicious user and erasures caused by node failures. Our analysis of the Low Rank Parity Check codes (LRPC) has shown that they are more suited to wireless sensor networks and they perform better than Gabidulin codes used in the literature. This analysis has been generalized in the multisource context and has shown that LRPC codes are more efficient in this context compared to Gabidulin codes. These contributions give a new incentive for the use of rank metric codes and offer interesting perspectives.

A dissertation submitted in ful lment of the requirements for the degree of Masters in Science in the Information Engineering School of Electrical and Information Engineering August 2016 / The work presented extends and contributes to research in error-control coding and information theory. The work focuses on the construction of block codes using an IIR lter structure. Although previous works in this area uses FIR lter structures for error-detection, it was inherently used in conjunction with other error-control codes, there has not been an investigation into using IIR lter structures to create codewords, let alone to justify its validity. In the research presented, linear block codes are created using IIR lters, and the error-correcting capabilities are investigated. The construction of short codes that achieve the Griesmer bound are shown. The potential to construct long codes are discussed and how the construction is constrained due to high computational complexity is shown. The G-matrices for these codes are also obtained from a computer search, which is shown to not have a Quasi-Cyclic structure, and these codewords have been tested to show that they are not cyclic. Further analysis has shown that IIR lter structures implements truncated cyclic codes, which are shown to be implementable using an FIR lter. The research also shows that the codewords created from IIR lter structures are valid by decoding using an existing iterative soft-decision decoder. This represents a unique and valuable contribution to the eld of error-control coding and information theory. / MT2017

Thesis (Master of Engineering in Electrical Engineering)--Cape Peninsula University of Technology, 2018. / This thesis will focus on preventing and overcoming the effects of radiation in RAM on board the ZA cube 2 nanosatellite. The main objective is to design, implement and test an effective error detection and correction (EDAC) system for nanosatellite applications using a SoC development board. By conducting an in-depth literature review, all aspects of single-event effects are investigated, from space radiation right up to the implementation of an EDAC system. During this study, Hamming code was identified as a suitable EDAC scheme for the prevention of single-event effects. During the course of this thesis, a detailed radiation study of ZA cube 2’s space environment is conducted. This provides insight into the environment to which the satellite will be exposed to during orbit. It also provides insight which will allow accurate testing should accelerator tests with protons and heavy ions be necessary. In order to understand space radiation, a radiation study using ZA cube 2’s orbital parameters was conducted using OMERE and TRIM software. This study included earth’s radiation belts, galactic cosmic radiation, solar particle events and shielding. The results confirm that there is a need for mitigation techniques that are capable of EDAC. A detailed look at different EDAC schemes, together with a code comparison study was conducted. There are two types of error correction codes, namely error detection codes and error correction codes. For protection against radiation, nanosatellites use error correction codes like Hamming, Hadamard, Repetition, Four Dimensional Parity, Golay, BCH and Reed Solomon codes. Using detection capabilities, correction capabilities, code rate and bit overhead each EDAC scheme is evaluated and compared. This study provides the reader with a good understanding of all common EDAC schemes. The field of nanosatellites is constantly evolving and growing at a very fast speed. This creates a growing demand for more advanced and reliable EDAC systems that are capable of protecting all memory aspects of satellites. Hamming codes are extensively studied and implemented using different approaches, languages and software. After testing three variations of Hamming codes, in both Matlab and VHDL, the final and most effective version was Hamming [16, 11, 4]2. This code guarantees single error correction and double error detection. All developed Hamming codes are suited for FPGA implementation, for which they are tested thoroughly using simulation software and optimised.

by Tse, Yan Tung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 75-76). / Abstracts in English and Chinese. / Abstract --- p.i / 摘要 --- p.i / Acknowledgements --- p.ii / Table of Contents --- p.iii / List of Figures --- p.vi / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Types of 2D Barcodes --- p.3 / Chapter 1.2 --- Reading 2D Barcodes --- p.5 / Chapter 1.3 --- Thesis Organization --- p.8 / Chapter Chapter 2 --- Related Works --- p.9 / Chapter 2.1 --- DataMatrix --- p.9 / Chapter 2.2 --- Original MaxiCode --- p.11 / Chapter 2.3 --- Spatial Methods for MaxiCode --- p.12 / Chapter 2.4 --- Summary --- p.14 / Chapter Chapter 3 --- Reading 2D Barcode on Uneven Surfaces --- p.15 / Chapter 3.1 --- The Image Processing Framework --- p.15 / Chapter 3.2 --- The Scanning Environment --- p.17 / Chapter 3.3 --- Perspective Transform --- p.20 / Chapter Chapter 4 --- Uneven Surface Models --- p.23 / Chapter 4.1 --- Cylindrical Surfaces --- p.24 / Chapter 4.2 --- General Uneven Surfaces --- p.26 / Chapter Chapter 5 --- """Patch-wise"" Barcode Reading" --- p.28 / Chapter 5.1 --- The Inputs --- p.28 / Chapter 5.2 --- The Registration Process --- p.29 / Chapter 5.3 --- Patch Cutting --- p.33 / Chapter Chapter 6 --- Registering Cells in a Patch --- p.37 / Chapter 6.1 --- Document Skew Detection: Projection Profiles --- p.38 / Chapter 6.2 --- Radon Transform Based Orientation Detection --- p.41 / Chapter 6.3 --- Identifying Row/column Boundaries --- p.45 / Chapter 6.4 --- Detecting Cell Width --- p.50 / Chapter 6.5 --- Calculating Transform Parameters --- p.53 / Chapter Chapter 7 --- Patch Registration --- p.57 / Chapter 7.1 --- Matching Adjacent patches --- p.57 / Chapter 7.2 --- Expanding to the Entire Code Area --- p.60 / Chapter Chapter 8 --- Simulation and Results --- p.61 / Chapter 8.1 --- Implementation Details --- p.61 / Chapter 8.2 --- Comparison Methods --- p.63 / Chapter 8.3 --- Results --- p.63 / Chapter 8.4 --- Computation Costs --- p.68 / Chapter Chapter 9 --- Conclusion and Further Works --- p.73 / Bibliography --- p.75

"Experts in information theory have long been interested in the maximal size, A(n, d), of a binary error-correcting code of length n and minimum distance d, The problem of determining A(n, d) involves both the construction of good codes and the search for good upper bounds. For quite some time now, Delsarte's linear programming approach has been the dominant approach to obtaining the strongest general purpose upper bounds on the efficiency of error-correcting codes. From 1973 forward, the linear programming bound found many applications, but there were few significant theoretical advances until Schrijver proposed a new code upper bound via semidefinite programming in 2003. Using the Terwilliger algebra, a recently introduced extension of the Bose-Mesner algebra, Schrijver formulated a new SDP strengthening of the LP approach. In this project we look at the dual solutions of the semidefinite programming bound for binary error-correcting codes. We explore the combinatorial meaning of these variables for small n and d, such as n = 4 and d = 2. To obtain information like this, we wrote a computer program with both Matlab and CVX modules to get solution of our primal SDP formulation. Our program efficiently generates the primal solutions with corresponding constraints for any n and d. We also wrote a program in C++ to parse the output of the primal SDP problem, and another Matlab script to generate the dual SDP problem, which could be used in assigning combinatorial meaning to the values given in the dual optimal solution. Our code not only computes both the primal and dual optimal variable values, but allows the researcher to display them in meaningful ways and to explore their relationship and dependence on arameters. These values are expected to be useful for later study of the combinatorial meaning of such solutions."

Based on the weight properties of network codes, we present the refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for linear network codes. We give two different algorithms to construct network codes with minimum distance constraints, both of which can achieve the refined Singleton bound. The first algorithm finds a codebook based on a given set of local encoding kernels defining a linear network code. The second algorithm finds a set of of local encoding kernels based on a given classical error-correcting code satisfying a certain minimum distance requirement. / First, the error correction/detection correction capabilities of a network code is completely characterized by a parameter which is equivalent to the minimum Hamming distance when the network code is linear and the weight measure on the error vectors is the Hamming weight. Our results imply that for a linear network code with the Hamming weight being the weight measure on the error vectors, the capability of the code is fully characterized by a single minimum distance. By contrast, for a nonlinear network code, two different minimum distances are needed for characterizing the capabilities of the code for error correction and for error detection. This leads to the surprising discovery that for a nonlinear network code, the number of correctable errors can be more than half of the number of detectable errors. (For classical algebraic codes, the number of correctable errors is always the largest integer not greater than half of the number of detectable errors.) / Network error correction provides a new method to correct errors in network communications by extending the strength of classical error-correcting codes from a point-to-point model to networks. This thesis considers a number of fundamental problems in coherent network error correction. / We further define equivalence classes of weight measures with respect to a general channel model. Specifically, for any given channel, the minimum weight decoders for two different weight measures are equivalent if the two weight measures belong to the same equivalence class. In the special case of network coding, we study four weight measures and show that all the four weight measures are in the same equivalent class for linear network codes. Hence they are all equivalent for error correction and detection when applying minimum weight decoding. / Yang, Shenghao. / Adviser: Raymond W.H. Yeung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3708. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 89-93). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

This dissertation explores the drivers of why organizations, as members of its market category, choose to conform or deviate from the category’s codes. In essence, codes are the social rules category members are expected to abide by and that underpin the very existence of a category. Given the importance of producer conformity in upholding a category’s continued existence, code deviance then seems a counterintuitive strategy to pursue. Nonetheless, organizations are known to defy codes in certain instances, sometimes even pairing the violation simultaneously with conformity to other codes. On top of this, organizations also seem to be able to strategically decide which codes they will abide by to a certain extent. Each of the three papers in this dissertation investigates why organizations may choose to either conform or deviate by, respectively, examining (1) the identity difference between the code violator with the potential adopter of the code violation, (2) the taken-for-grantedness of the category the organization is a part of, and (3) the individual status and organizational identity (insider-outsider) of the producer. The main overarching finding of this dissertation is that organizations will take into account both its internal resources and external socio-environment to decide which strategy it will deploy and whether it can afford to do so. All in all, this dissertation specifies how the three factors mentioned may affect an organization’s propensity to conform or violate to category codes.