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hill r first course coding theory 1990

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Textbook in PDF format Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study. The aim of this book is to provide an elementary treatment of the theory of error-correcting codes, assuming no more than high school mathematics and the ability to carry out matrix arithmetic. The book is intended to serve as a self-contained course for second or third year mathematics undergraduates, or as a readable introduction to the mathematical aspects of coding for students in engineering or computer science. Notation Introduction to error-correcting codes Introduction to error-correcting codes The transmission of photographs from deep-space Exercises 1 The main coding theory problem The main coding theory problem Equivalence of codes Binomial coefficients Perfect codes Balanced block designs Concluding remarks on Chapter 2 Exercises 2 An introduction to finite fields An introduction to finite fields The ISBN code Exercises 3 Vector spaces over finite fields Vector spaces over finite fields Exercises 4 Introduction to linear codes Introduction to linear codes Equivalence of linear codes Exercises 5 Encoding and decoding with a linear code Encoding with a linear code Decoding with a linear code Probability of error correction Symbol error rate Probability of error detection Concluding remark on Chapter 6 Exercises 6 The dual code, the parity-check matrix, and syndrome decoding The dual code and the parity-check matrix Syndrome decoding Incomplete decoding Exercises 7 The Hamming codes The Hamming codes Decoding with a binary Hamming code Extended binary Hamming codes A fundamental theorem q-ary Hamming codes Decoding with a q-ary Hamming code Shortening a code Concluding remarks on Chapter 8 Exercises 8 Perfect codes Perfect codes The binary Golay [23, 12, 7]-code The ternary Golay [11, 6, 5]-code Are there any more perfect codes? t-designs Remaining problems on perfect codes Concluding remarks Exercises 9 Codes and Latin squares Latin squares Mutually orthogonal Latin squares Optimal single-error-correcting codes of length 4 Sets of t mutually orthogonal Latin squares Exercises 10 A double-error correcting decimal code and an introduction to BCH codes A double-error correcting decimal code and an introduction to BCH codes Some preliminary results from linear algebra A double-error-correcting modulus 11 code A class of BCH codes Outline of the error-correction procedure (assuming ≤ t errors) Concluding remarks Exercises 11 Cyclic codes Cyclic codes Polynomials The division algorithm for polynomials The ring of polynomials modulo f(x) The finite fields GF(p^h), h > 1 Back to cyclic codes The check polynomial and the parity-check matrix of a cyclic code The binary Golay code The ternary Golay code Hamming codes as cyclic codes Concluding remarks on Chapter 12 Exercises 12 Weight enumerators Weight enumerators Probability of undetected errors Exercises 13 The main linear coding theory problem The main linear coding theory problem The MLCT problem for d = 3 (or Hamming codes revisited) The projective geometry PG(r - 1, q) The MLCT problem for d = 4 The determination of max₃(3, q) The determination of max₃(4, q) for q odd The values of B_q(n, 4), for n ≤ q² + 1 Remarks on max₃(r, q) for r > 5 Concluding remarks on Chapter 14 Exercises 14 MDS codes MDS codes The known results concerning Conjecture 15.2 Concluding remarks on Chapter 15 Exercises 15 Concluding remarks, related topics, and further reading Concluding remarks, related topics, and further reading Burst error-correcting codes Convolutional codes Cryptographic codes Variable-length source codes Exercises 16 Solutions to exercises

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hill r first course coding theory 1990

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