Introduction to Coding Theory

Although its roots lie in information theory, the applications of coding theory now extend to statistics, cryptography, and many areas of pure mathematics, as well as pervading large parts of theoretical computer science, from universal hashing to numerical integration. Introduction to Coding Theory introduces the theory of error-correcting codes in a thorough but gentle presentation. Part I begins with basic concepts, then builds from binary linear codes and Reed-Solomon codes to universal hashing, asymptotic results, and 3-dimensional codes. Part II emphasizes cyclic codes, applications, and the geometric desciption of codes. The author takes a unique, more natural approach to cyclic codes that is not couched in ring theory but by virtue of its simplicity, leads to far-reaching generalizations. Throughout the book, his discussions are packed with applications that include, but reach well beyond, data transmission, with each one introduced as soon as the codes are developed. Although designed as an undergraduate text with myriad exercises, lists of key topics, and chapter summaries, Introduction to Coding Theory explores enough advanced topics to hold equal value as a graduate text and professional reference. Mastering the contents of this book brings a complete understanding of the theory of cyclic codes, including their various applications and the Euclidean algorithm decoding of BCH-codes, and carries readers to the level of the most recent research.

Part I: AN ELEMENTARY INTRODUCTION TO CODING THE CONCEPT OF CODING Bitstrings and Binary Operations The Hamming Distance Binary Codes Error-Correcting Codes in General The Binary Symmetric Channel The Sphere-Packing Bound BINARY LINEAR CODES The Concept of Binary Linear Codes Block Coding The Effect of Coding Duality Binary Hamming and Simplex Codes Principle of Duality GENERAL LINEAR CODES Prime Fields Finite Fields Linear Codes Over Finite Fields Duality and Orthogonal Arrays Weight Distribution The Game of Set Syndrome Decoding REED-SOLOMON CODES RECURSIVE CONSTRUCTION I UNIVERSAL HASHING DESIGNS AND THE BINARY GOLAY CODE SHANNON ENTROPY ASYMPTOTIC RESULTS 3-DIMENSIONAL CODES, PROJECTIVE PLANES SUMMARY AND OUTLOOK Part II: THE THEORY OF CODES AND THEIR APPLICATIONS SUBFIELD CODES AND TRACE CODES The Trace Trace Codes and Subfield Codes Galois Closed Codes Automorphism Groups CYCLIC CODES Some Primitive Cyclic Codes of Length 15 Theory of Cyclic Codes Decoding BCH-Codes Constacyclic Codes Remarks RECURSIVE CONSTRUCTIONS, COVERING RADIUS Construction X Covering Radius OA IN STATISTICS AND COMPUTER SCIENCE OA and Independent Random Variables Linear Shift Register Sequences Cryptography Two-Point Based Sampling Resilient Functions Derandomization of Algorithms Authentication and Universal Hashing THE GEOMETRIC DESCRIPTION OF CODES Linear Codes as Sets of Points Quadratic Forms, Bilinear Forms and Caps Caps: Constructions and Bounds ADDITIVE CODES Basic Constructions and Applications Additive Cyclic Codes Quantum Codes THE LAST CHAPTER The Linear-Programming Bound Algebraic-Geometric Codes List Decoding Expander Codes tms-Nets Sphere Packings Designs Nonlinear Codes REFERENCES