Algebraic Codes For Error Correction In Digital Communication Systems

Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible error-free in the presence of noise. Subsequently the notion of using error correcting codes to mitigate the effects of noise in digital transmission was introduced by R. Hamming. Algebraic codes, codes described using powerful tools from algebra took to the fore early on in the search for good error correcting codes. Many classes of algebraic codes now exist and are known to have the best properties of any known classes of codes. An error correcting code can be described by three of its most important properties length, dimension and minimum distance. Given codes with the same length and dimension, one with the largest minimum distance will provide better error correction. As a result the research focuses on finding improved codes with better minimum distances than any known codes. Algebraic geometry codes are obtained from curves. They are a culmination of years of research into algebraic codes and generalise most known algebraic codes. Additionally they have exceptional distance properties as their lengths become arbitrarily large. Algebraic geometry codes are studied in great detail with special attention given to their construction and decoding. The practical performance of these codes is evaluated and compared with previously known codes in different communication channels. Furthermore many new codes that have better minimum distance to the best known codes with the same length and dimension are presented from a generalised construction of algebraic geometry codes. Goppa codes are also an important class of algebraic codes. A construction of binary extended Goppa codes is generalised to codes with nonbinary alphabets and as a result many new codes are found. This construction is shown as an efficient way to extend another well known class of algebraic codes, BCH codes. A generic method of shortening codes whilst increasing the minimum distance is generalised. An analysis of this method reveals a close relationship with methods of extending codes. Some new codes from Goppa codes are found by exploiting this relationship. Finally an extension method for BCH codes is presented and this method is shown be as good as a well known method of extension in certain cases

Random matrix theory and the loss surfaces of neural networks

Neural network models are one of the most successful approaches to machine learning, enjoying an enormous amount of development and research over recent years and finding concrete real-world applications in almost any conceivable area of science, engineering and modern life in general. The theoretical understanding of neural networks trails significantly behind their practical success and the engineering heuristics that have grown up around them. Random matrix theory provides a rich framework of tools with which aspects of neural network phenomenology can be explored theoretically. In this thesis, we establish significant extensions of prior work using random matrix theory to understand and describe the loss surfaces of large neural networks, particularly generalising to different architectures. Informed by the historical applications of random matrix theory in physics and elsewhere, we establish the presence of local random matrix universality in real neural networks and then utilise this as a modeling assumption to derive powerful and novel results about the Hessians of neural network loss surfaces and their spectra. In addition to these major contributions, we make use of random matrix models for neural network loss surfaces to shed light on modern neural network training approaches and even to derive a novel and effective variant of a popular optimisation algorithm. Overall, this thesis provides important contributions to cement the place of random matrix theory in the theoretical study of modern neural networks, reveals some of the limits of existing approaches and begins the study of an entirely new role for random matrix theory in the theory of deep learning with important experimental discoveries and novel theoretical results based on local random matrix universality.Comment: 320 pages, PhD thesi

Auxiliary functions in transcendence proofs

We discuss the role of auxiliary functions in the development of transcendental number theory

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book