Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes

This paper presents a method for achieving equilibrium in the ISING Hamiltonian when confronted with unevenly distributed charges on an irregular grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our approach involves dimensionally expanding the system, substituting charges with circulants, and representing distances through circulant shifts. This results in a systematic mapping of the charge system onto a space, transforming the irregular grid into a uniform configuration, applicable to Torical and Circular Hyperboloid Topologies. The paper covers fundamental definitions and notations related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine. It explores the marginalization problem in code on the graph probabilistic models for evaluating the partition function, encompassing exact and approximate estimation techniques. Rigorous proof is provided for the attainability of equilibrium states for the Boltzmann machine under Torical and Circular Hyperboloid, paving the way for the application of our methodology. Practical applications of our approach are investigated in Finite Geometry QC-LDPC Codes, specifically in Material Science. The paper further explores its effectiveness in the realm of Natural Language Processing Transformer Deep Neural Networks, examining Generalized Repeat Accumulate Codes, Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium method is showcased across diverse scientific domains without the use of specific section delineations.Comment: 16 pages, 29 figures. arXiv admin note: text overlap with arXiv:2307.1577

Decoding Cyclic Codes up to a New Bound on the Minimum Distance

A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula

On products and powers of linear codes under componentwise multiplication

In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that could otherwise probably not fit in a regular research paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few open problems. Our presentation alternates between two points of view, which the theory intertwines in an essential way: that of combinatorial coding, and that of algebraic geometry. In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, for a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected in the final "open questions" sectio

Contemporary Coding Theory

Coding Theory naturally lies at the intersection of a large number of disciplines in pure and applied mathematics. A multitude of methods and means has been designed to construct, analyze, and decode the resulting codes for communication. This has suggested to bring together researchers in a variety of disciplines within Mathematics, Computer Science, and Electrical Engineering, in order to cross-fertilize generation of new ideas and force global advancement of the field. Areas to be covered are Network Coding, Subspace Designs, General Algebraic Coding Theory, Distributed Storage and Private Information Retrieval (PIR), as well as Code-Based Cryptography

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

Performance evaluation of low-density parity-check codes

LDPC codes were first introduced by Robert Gallager in 1960. Due to the complexity of the codes and the limitations of the then rudimentary computer resources the codes were neglected as a viable form of FEC. LDPC codes were rediscovered by Tanner in 1981 when he generalized the codes and provided a means of graphical representation of LDPC codes. LDPC codes were again neglected until the work of MacKay et al in the mid to late 1990’s resurrected interest in the codes when they were discovered to out perform the then premium Turbo codes. This dissertation specifically describes the process of encoding and decoding LDPC codes and demonstrates the performance comparison between the various types of decoders in terms of bit error rate performance factors

A study of major coding techniques for digital communication Final report

Coding techniques for digital communication channel