Easily decoded error-correcting codes and techniques for their generation

Imperial Users onl

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Z₄-linear codes

Coding theory is a branch of mathematics that began in the 1940\u27s to correct errors caused by noise in communication channels. It is know that certain nonlinear codes satisfy the MacWilliams Identity. Much research has been done to explain this relation- ship. In the early 1990\u27s, five coding theorists discovered that nonlinear codes have linear properties if viewed under the alphabet Z4 rather than the usual alphabet F2. In 1994, these coding theorists published their results in a joint paper in IEEE Transactions on Information Theory. In this study, linear codes and nonlinear codes are introduced and characterized as Z4-linear codes to understand the importance of this discovery

Algebraic curves and applications to coding theory.

by Yan Cho Hung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 122-124).Abstract also in Chinese.Chapter 1 --- Complex algebraic curves --- p.6Chapter 1.1 --- Foundations --- p.6Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9Chapter 1.1.3 --- Complex projective curves in P2 --- p.11Chapter 1.1.4 --- Affine and projective curves --- p.13Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16Chapter 1.2.1 --- Intersection multiplicity --- p.16Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18Chapter 1.2.3 --- Cubic curves --- p.21Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23Chapter 1.4 --- Riemann surfaces --- p.26Chapter 1.4.1 --- Weierstrass &-function --- p.26Chapter 1.4.2 --- Riemann surfaces and examples --- p.27Chapter 1.5 --- Differentials on Riemann surfaces --- p.28Chapter 1.5.1 --- Holomorphic differentials --- p.28Chapter 1.5.2 --- Abel's Theorem for tori --- p.31Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32Chapter 1.6 --- Singular curves --- p.36Chapter 1.6.1 --- Resolution of singularities --- p.37Chapter 1.6.2 --- The topology of singular curves --- p.45Chapter 2 --- Coding theory --- p.48Chapter 2.1 --- An introduction to codes --- p.48Chapter 2.1.1 --- Efficient noiseless coding --- p.51Chapter 2.1.2 --- The main coding theory problem --- p.56Chapter 2.2 --- Linear codes --- p.58Chapter 2.2.1 --- Syndrome decoding --- p.63Chapter 2.2.2 --- Equivalence of codes --- p.65Chapter 2.2.3 --- An introduction to cyclic codes --- p.67Chapter 2.3 --- Special linear codes --- p.71Chapter 2.3.1 --- Hamming codes --- p.71Chapter 2.3.2 --- Simplex codes --- p.72Chapter 2.3.3 --- Reed-Muller codes --- p.73Chapter 2.3.4 --- BCH codes --- p.75Chapter 2.4 --- Bounds on codes --- p.77Chapter 2.4.1 --- Spheres in Zn --- p.77Chapter 2.4.2 --- Perfect codes --- p.78Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85Chapter 3.1 --- Algebraic curves over finite fields --- p.85Chapter 3.1.1 --- Affine varieties --- p.85Chapter 3.1.2 --- Projective varieties --- p.37Chapter 3.1.3 --- Morphisms --- p.89Chapter 3.1.4 --- Rational maps --- p.91Chapter 3.1.5 --- Non-singular varieties --- p.92Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93Chapter 3.2 --- Goppa codes --- p.96Chapter 3.2.1 --- Elementary Goppa codes --- p.96Chapter 3.2.2 --- The affine and projective lines --- p.98Chapter 3.2.3 --- Goppa codes on the projective line --- p.102Chapter 3.2.4 --- Differentials and divisors --- p.105Chapter 3.2.5 --- Algebraic geometric codes --- p.112Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116Bibliograph

Trellis Decoding And Applications For Quantum Error Correction

Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost. Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products. Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.Ph.D

Private Information Retrieval: Combinatorics of the Star-Product Scheme

In coded private information retrieval (PIR), a user wants to download a file from a distributed storage system without revealing the identity of the file. We consider the setting where certain subsets of servers collude to deduce the identity of the requested file. These subsets form an abstract simplicial complex called the collusion pattern. In this thesis, we study the combinatorics of the general star-product scheme for PIR under the assumption that the distributed storage system is encoded using a repetition code

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

A Combinatorial Commutative Algebra Approach to Complete Decoding

Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí