Error-Correcting Codes and Finite Fields

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ISBN: 9780192690678 | 0192690671
Cover: Paperback
Copyright: 8/29/1996

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This textbook is a reprint of Chapters 1-20 of the original hardback edition. It provides the reader with the tools necessary to implement modern error-processing schemes. The material on algebraic geometry and geometric Goppa codes, which is not part of a standard introductory course oncoding theory, has been omitted.The book assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples which motivate and explain the theory.The book is in four parts. The first introduces the basic ideas of coding theory. The second and third cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. These parts are linked by their uses of Eulid's algorithm as a central technique. The fourth parttreats classical Goppa codes.

PART 1 BASIC CODING THEORY

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1 Introduction Errors of transmission. Examples from natural language. Channel models. The binary symmetric channel. Three simple codes (a parity check code, a triple repetition code, and a triple parity check code).

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2 Block codes, weight, and distance Block codes. Block length, message block length, and rate. Definition of Hamming Weight and distance. Minimum distance, error detection, and error correction. Block and message success probabilities. Calculation of error detection/correction probabilities for the examples of Chapter 1. Discussion of Shannon's theorem (without proof).

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3 Linear codes Definition of linear codes and fields. Dimension and rate. The generator matrix. Standard form generator matrices and systematic encoding. Message and check bits. The check matrix. Uniqueness of standard form generator and check matrices.

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4 Error processing for linear codes Decoding by cosets (standard array). Coset leaders and syndromes. Code can correct single errors if and only if check matrix has distinct non-Zero columns. Conditions for multiple error correction.

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5 Hamming codes and the binary Golay codes Definition of the sequence of binary Hamming codes Ham(k) by their check matrices. Success probabilities for Hamming codes. Long Hamming codes are very efficient, but poor at correcting errors. Perfect codes. Construction of the binary Golay codes by Turyn's method.

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Appendix LA Linear algebra The laws of arithmetic: rings, domains, and fields. Elementary vector space theory. Bases and dimension. Elementary matrix theory. Row operations, rank, and nullity. Vandermonde matrices

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PART 2 FINITE FIELDS

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6 Introduction and an example The need for fields other than Z/2. An attempt to construct a field of order 16. Z/16 will not do. Polynomial arithmetic. Table of GF(16).

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7 Euclid's algorithm Division with remainder. Euclidean domains with F[x] and Z as examples. Euclid's algorithm in tabular form for a Euclidean domain. Finding the highest common factor in the form (a,b) = ua + vb. Extras. Relations between entries in the table for Euclid's algorithm. Continued fractions. Convergents, the entries in the tabular form of Euclid's algorithm and convergents to continued fractions.

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8 Invertible and irreducible elements Definition of invertible elements in a Euclidean domain. Definition of irreducible elements in a Euclidean domain. The 1-trick and the key property of irreducible elements. Discussion of unique factorization. Extras. Proof of unique factorization.

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9 The Construction of fields Construction of the Factor ring (residue class ring) D/a. D/a is a field if and only if a is irreducible. Using Euclid's algorithm to perform field arithmetic in F[x]/f(x). Examples: GF(16) as GF(2) [x]/(x4 + x3 + 1), Z/787.

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10 The structure of finite fields The prime field and the characteristic. The order of a finite field. The frobenius automorphism x-> xp. Fermat's little theorem: if F has order q then all its elements are roots of xq - x. Example: GF(16).

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11 Roots of polynomials The evaluation map. Its basic properties (i.e. it is a homomorphism). The formal derivative. Horner's scheme for evaluating a polynomial. Extension of Horner's scheme to evaluate the derivative. Multiple roots. The minimal polynomial of x. Characterization of the minimal polynomial and the set (ideal) of polynomials with x as a root. List of minimal polynomials of elements of GF(16). Isomorphism F[x] = F[x]/mpa(x). Construction of a field contining a root of a given polynomial. Existence of finite fields of all legal orders. Extras. Calculation of the minimum polynomial of B using the Frobenius automorphism.

166

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12 Primitive elements Definition of primitive elements. Primitive elements of GF(16). Logarithms for calculating products and quotients in finite fields. Zech logarithms for calculating sums. Primitive polynomials. Existence of primitive elements. Existence of subfields of all legal orders. Isomorphism of fields of the same order. The polynomial xq - x is the product of all irreducible polynomials of degree dividing q. Extras. The number of irreducible polynomials of given degree.

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Appendix PF Polynomials over a field Recapitulation of the basic theory of polynomials over a field. Definition, addition, multiplication, degree. F[x] is an integral domain. Division with remainder. Polynomials in two indeterminates.

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PART 3 BCH CODES AND OTHER POLYNOMIAL CODES

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13 BCH codes as subcodes of Hamming codes Example: BCH(4,2) constructed from Ham(4) by extending the check matrix H4. Extensions must not be linear (or quadratic). View Hk as having entries in GF(2k). Criterion for multiple error correction. Vandermonde matrices. The full check matrix V4, 2 and the reduced check matrix H4, 2 (Vk, t and Hk, t in general). Example BCH(4, 3). BCH(k, t) can correct t errors per block. It has block lenght 2k - 1 and dimension > 2k - 1 - kt.

201

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14 BCH codes as polynomial codes Example: BCH(4, 3) used throughtout to illustrate the theory. Code words as polynomials. Redefine BHC (k, t) in terns of polynomials. The generator polynomial of BCH (k, t). Dimension of BCH(k, t). Encoding by multipleication. The check polynomial of BCH(k, t). Use of the check polynomial, to Verify and decode a code word. Systematic encoding by division with remainder. Extras. Polynomial codes in general. Cyclic codes in general. Recognition of Polynomial and Cyclic Codes.

216

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15 BCH error correction: (1) the fundamental equation Example BCH(4, 3) continued. The Error Polynomial and error locations. Syndromes; calculation via Horner's scheme. Direct solution of case of two errors. The Syndrome polynomial. Derivation of the fundamental equation. The error locator, error evaluator and error co-evaluator polynomials. Uniqueness of these as solutions of the fundamental equation.

233

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16 BCH error correction: (2) an algorithm Example BCH(4, 3) continued. The Sugiyama-Kasahara-Hirasawa-Namekawa error processor using Euclid's algorithm. Failure modes of the algorithm.

249

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17 Reed-Solomon codes and burst error correction Example RS(4, 3) used throughout. The Reed-Solomon code RS(k, t) corresponding to BCH(k, t). Adaptation of the decoding algorithm to RS(k, t). Failure modes. RS(k, t) as a cyclic code over GF(2k). Parameters of RS(k, t) over GF(2k) and GF(2). RS(k, t) as a burst error-correcting code. Comparison with interleaved BCH(k, t). Extras. Detailed proofs of the statements concerning error modes.

267

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18 Bounds on codes Extending, shortening and puncturing a code. The Singleton bound. MDS codes. Reed-Solomon codes are MDS. Coding bounds based on sphere packing: the Hamming bound, the Gibert-Varshamov bound. The asymptotic Gilbert-Varshamov bound. Good and bad families of codes. BCH codes are bad in relation to their designed distance, although their parameters for moderate block lengths are good. Estimates for the true minimum distance. Discussion of the fact that BCH code are still bad for their true minimum distance. Extras. Proof of the estimates used in establishing the asymptotic Gilbert-Varshamov bound.

287

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PART 4 CLASSICAL GOPPA CODES

303

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19 Classical Goppa codes Definition of the Goppa Code GC(p, g) with Goppa Polynomial g(x). Rational functions over GF(q). Dimension of GC(P, g), special case of binary Goppa codes. Minimum distance of the GC(p, g). Goppa codes and codes of BCH-type.

303

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20 Classical Goppa codes: error processing The error locator and error evaluator polynomial, the fundamental equation. Euclid's algorithm decoding for GC(p, g). Extras. Classical Goppa codes are bad for their designed distance, but there exists a sequence of classical Goppa codes that is good for the true minimum distance.

320

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Bibliography

333

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Index

337

Bibliography

333

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Index

337

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Error-Correcting Codes and Finite Fields

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