Includes bibliographical references and index.

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Front Cover; Codes and Rings; Copyright; Contents; Foreword; Introduction; 1 Motivation; 1.1 The Geometry of Codes; 1.2 Sequences; 1.2.1 Periodic Correlation; 1.3 Lattices; 1.4 Maps; 1.4.1 Prehistory; 1.4.2 History; 1.4.3 Present; 1.5 Designs; References; 2 Rings; 2.1 Basic Rings; 2.2 Local Rings; 2.3 Galois Rings; 2.3.1 Hensel Lifting; 2.3.2 Bottom-up; 2.3.3 Top-down; 2.3.4 Multiplicative Structure; 2.4 Skew Polynomial Rings; 2.5 Chain Rings; 2.6 Frobenius Rings; References; 3 Distances; 3.1 The Lee Metric; 3.1.1 The Sphere-packing Bound; 3.1.2 A Plotkin-like Bound

3.1.3 A Singleton-like Bound3.1.4 Other Bounds; 3.2 The Homogeneous Metric; 3.2.1 Sphere-packing Bound; 3.2.1.1 A Plotkin-like Bound; 3.2.1.2 A Singleton-like Bound; 3.2.1.3 Other Bounds; 3.3 Hamming Metric; 3.3.1 Codes over Frobenius Rings; 3.3.2 A Griesmer-like Bound; 3.3.3 A Singleton-like Bound; References; 4 Few Weight Codes; 4.1 One-weight Codes; 4.1.1 Preliminaries; 4.1.2 One-homogeneous Weight Codes over Finite Chain Rings; 4.2 Two-weight Codes; 4.2.1 Linear Codes and Geometries over Finite Frobenius Rings; 4.2.2 Two-weight Codes over a Finite Frobenius Ring

4.2.3 Two-weight Codes and Strongly Regular Graphs4.2.4 Constructions; 4.2.5 Properties of Codes with Two Homogeneous Weights; 4.2.6 Gray Isometries; 4.3 On Two-weight Z2k-codes; 4.3.1 Background; 4.3.1.1 Graph Theory; 4.3.2 Coding Theory; 4.3.3 Z2k-codes and Syndrome Graphs; 4.3.4 Two-weight Z4-codes; References; 5 Linear Codes; 5.1 Chain Rings; 5.1.1 Generator Matrix; 5.1.2 Dual Code; 5.1.3 Free Codes; 5.2 Modular Independence; References; 6 Self-dual Codes; 6.1 Chain Rings; 6.1.1 Existence Conditions; 6.1.2 Type II Codes; 6.2 Commutative Frobenius Rings; 6.2.1 CRT Theory

6.2.2 Existence Results6.3 Noncommutative Frobenius Rings; References; 7 Cyclic Codes; 7.1 Splitting Codes; 7.1.1 Divisors of xn-1; 7.1.2 A Characterization of Splitting Cyclic Codes; 7.2 Polycyclic Codes; 7.2.1 Sequential Codes; 7.3 Multivariable Codes; 7.3.1 Multivariable Semisimple Codes; 7.3.1.1 Decomposition of R[X1,...,Xr]/; 7.3.1.2 Description of the Codes; 7.3.1.3 Hamming Distance of the Codes; 7.3.2 Dual Codes of Abelian Semisimple Codes; 7.3.3 Self-dual Abelian Semisimple Codes; References; 8 Quasicyclic Codes; 8.1 Quasicyclic Codes over Finite Fields

Codes and Rings: Theory and Practice is a systematic review of literature that focuses on codes over rings and rings acting on codes. Since the breakthrough works on quaternary codes in the 1990s, two decades of research have moved the field far beyond its original periphery. This book fills this gap by consolidating results scattered in the literature, addressing classical as well as applied aspects of rings and coding theory. New research covered by the book encompasses skew cyclic codes, decomposition theory of quasi-cyclic codes and related codes and duality over Frobenius rings. Primarily suitable for ring theorists at PhD level engaged in application research and coding theorists interested in algebraic foundations, the work is also valuable to computational scientists and working cryptologists in the area.

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