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Mathematics of Coding Theory Information, Compression, Error Correction, and Finite Fields

ISBN-10: 0131019678
ISBN-13: 9780131019676
Edition: 2004
Authors: Paul Garrett
List price: $95.33
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Description: For undergraduate courses in Coding Theory and Network Security, in departments of Mathematics, Electrical Engineering, and Computer Science. This text makes a very accessible introduction to a very important contemporary application of number  More...

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Book details

List price: $95.33
Copyright year: 2004
Publisher: Prentice Hall PTR
Publication date: 11/19/2003
Binding: Hardcover
Pages: 408
Size: 7.25" wide x 9.75" long x 1.00" tall
Weight: 1.694
Language: English

For undergraduate courses in Coding Theory and Network Security, in departments of Mathematics, Electrical Engineering, and Computer Science. This text makes a very accessible introduction to a very important contemporary application of number theory, abstract algebra, and probability. From credit cards to cellphones, codes are discussed both mathematically and with regard to industry protocols.

Probability
Counting
Preliminary Ideas of Probability
More Formal View of Probability
Random Variables, Expected Values, Variance
Markov Inequality, Chebycheff Inequality
Law of Large Numbers
Information and Entropy
Uncertainty, Acquisition of Information
Entropy
Uniquely-Decipherable and Prefix Codes
Kraft and Macmillan Inequalities
Noiseless Coding
Noiseless Coding Theorem
Huffman Coding
Noisy Coding
Noisy channels
Example: Parity Checks
Decoding from a Noisy Channel
Channel Capacity
Noisy Coding Theorem
Cyclic Redundancy Checks
The Finite Field GF(2) with 2 Elements
Polynomials over GF(2)
Cyclic Redundancy Checks (CRC's)
What Errors Does a CRC Catch?
6
Reduction Algorithm
Divisibility
Factorization into Primes
Euclidean Algorithm
Integers Modulo M
The Finite Field Z/P for P Prime
Fermat's Little Theorem
Primitive Roots
Euler's Criterion
Fast Modular Exponentiation
Finite Fields
Making Fields
Examples of Field Extensions
Addition Modulo P
Multiplication Modulo P
Multiplicative Inverses Modulo P
Primitive Roots
Polynomials
Polynomials with Coefficients in a Field
Divisibility
Factoring and Irreducibility
Euclidean Algorithm
Unique Factorization
Introduction to Linear Codes
An Ugly Example
The Hamming Binary [7,4] Code
Some Linear Algebra
A Review of Row Reduction
Definition: Linear Codes
Syndrome Decoding
Berlekamp's Algorithm
Bounds for Codes
Hamming (Sphere-Packing) Bound
Gilbert-Varshamov Bound
Singleton Bound
Cyclic Codes
Minimum Distance in Linear Codes
Cyclic Codes
Primitive Roots
Characteristics of Fields
Multiple Factors in Polynomials
Cyclotomic Polynomials
Primitive Roots in Finite Fields
Primitive Roots Modulo Prime Powers
Counting Primitive Roots
Non-Existence
An Algorithm to Find Primitive Roots
Primitive Polynomials
Definitions
Examples Modulo 2
Testing for Primitivity
Example: Periods of LFSR's
Example: Two-Bit Errors Detected by CRC's
Basic Linear Codes
Vandermonde Determinants
More Check Matrices for Cyclic Codes
RS Codes
Hamming Codes (Again)
BCH Codes
Decoding BCH Codes
Concatenated Codes
Mirage Codes
Concatenated Codes
Justesen Codes
Some Explicit Irreducible Polynomials
Curves and Codes
Plane Curves
Singularities of Curves
Projective Plane Curves
Curves in Higher Dimensions
Genus, Divisors, Linear Systems
Geometric Goppa Codes
Tsfasman-Vladut-Zink Bound
Appendices
Sets and functions
Equivalence Relations
Stirling's Formula
Bibliography
Index

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