New Approaches to the Analysis and Design of Reed-Solomon Related Codes

The research that led to this thesis was inspired by Sudan's breakthrough that demonstrated that Reed-Solomon codes can correct more errors than previously thought. This breakthrough can render the current state-of-the-art Reed-Solomon decoders obsolete. Much of the importance of Reed-Solomon codes stems from their ubiquity and utility. This thesis takes a few steps toward a deeper understanding of Reed-Solomon codes as well as toward the design of efficient algorithms for decoding them. After studying the binary images of Reed-Solomon codes, we proceeded to analyze their performance under optimum decoding. Moreover, we investigated the performance of Reed-Solomon codes in network scenarios when the code is shared by many users or applications. We proved that Reed-Solomon codes have many more desirable properties. Algebraic soft decoding of Reed-Solomon codes is a class of algorithms that was stirred by Sudan's breakthrough. We developed a mathematical model for algebraic soft decoding. By designing Reed-Solomon decoding algorithms, we showed that algebraic soft decoding can indeed approach the ultimate performance limits of Reed-Solomon codes. We then shifted our attention to products of Reed-Solomon codes. We analyzed the performance of linear product codes in general and Reed-Solomon product codes in particular. Motivated by these results we designed a number of algorithms, based on Sudan's breakthrough, for decoding Reed-Solomon product codes. Lastly, we tackled the problem of analyzing the performance of sphere decoding of lattice codes and linear codes, e.g., Reed-Solomon codes, with an eye on the tradeoff between performance and complexity.</p

Quantum stabilizer codes and beyond

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. 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. 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. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing 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.Comment: Ph.D. Dissertation, Texas A&M University, 200

Partitions of codes

In this thesis we look at coding theory wherein we introduce the concept of perspective, a generalisation on the minimum distance of a code, which naturally leads to a partition of the code. Subsequently we introduce focused splittings, which shall be shown to be a generalisation of perfect codes. We investigate the existence of such objects, and address questions such as the complexity of finding a focused splittings, which we show to be NPComplete. We analyse the symmetries of focused splittings. We use focused splittings to address the problem of error correction and we construct an encoding method based on them. Finally we test this construction for various classes of focused splittings

Transmission and Coding of Information

2018/201

Algoritmos eficientes de búsqueda de códigos cíclicos y cíclicos acortados correctores de ráfagas de errores

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Informática, Departamento de Ingeniería del Software e Inteligencia Artificial, leída el 30/01/2013Depto. de Ingeniería de Software e Inteligencia Artificial (ISIA)Fac. de InformáticaTRUEUniversidad Complutense de MadridAgencia Española de Cooperación Internacional para el Desarrollo (AECID)unpu