Chapter 1
Introduction
Claude Shannon's 1948 paper "A Mathematical Theory of Communication" gave birth to the twin disciplines of information theory and coding theory. The basic goal is efficient and
Chapter 9
Weight and Distance
Enumeration
The weight and distance enumerators record the weight and distance information for the code. In turn they can be analyzed to reveal properties of the code.
Th
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A.2
A.2.1
Polynomial Algebra over Fields
Polynomial rings over fields
indeterminate polynomial
We have introduced fields in order to give arithmetic structure to our alphabet F . Our next wish i
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A.1
A.1.1
Basic Algebra
Fields
In doing coding theory it is advantageous for our alphabet to have a certain amount of mathematical structure. We are familiar at the bit level with boolean additi
Chapter 8
Cyclic Codes
Among the rst codes used practically were the cyclic codes which were generated using shift registers. It was quickly noticed by Prange that the class
of cyclic codes has a rich
Chapter 7
Codes over Subelds
In Chapter 6 we looked at various general methods for constructing new codes
from old codes. Here we concentrate on two more specialized techniques that
result from writin
Chapter 2
Sphere Packing and Shannon's Theorem
In the first section we discuss the basics of block coding on the m-ary symmetric channel. In the second section we see how the geometry of the codespace
Chapter 3
Linear Codes
In order to dene codes that we can encode and decode eciently, we add more
structure to the codespace. We shall be mainly interested in linear codes. A
linear code of length n o
Chapter 5
Generalized Reed-Solomon
Codes
In 1960, I.S. Reed and G. Solomon introduced a family of error-correcting codes
that are doubly blessed. The codes and their generalizations are useful in prac
Chapter 4
Hamming Codes
In the late 1940s Claude Shannon was developing information theory and coding as a mathematical model for communication. At the same time, Richard
Hamming, a colleague of Shann
Chapter 6
Modifying Codes
If one code is in some sense good, then we can hope to nd from it similar and
related codes that are also good. In this chapter we discuss some elementary
methods for modifyi
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A.3
A.3.1
Special Topics
The Euclidean algorithm
Let F be a field. In Theorem A.2.16 we gave a nonconstructive proof for the existence of the greatest common divisor of two polynomials a(x) and