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 reliable communication in an uncooperative (and possibl
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.
The most important result is MacWilliams Theorem, which w
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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 is then to give arithmetic structure to words formed fro
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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 addition (EXCLUSIVE OR) and multiplication (AND) within the s
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 algebraic structure, the rst indication that algebra
w
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 writing the eld F as a vector space over its subeld K. We wil
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 can be used to make coding judgements. This leads to t
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 over the eld F is a subspace of F n . Thus the words of
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 practice, and the mathematics that lies behind them is inte
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 Shannons at Bell Laboratories, found a need for error
correc
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 modifying a code in order to nd new codes. In two further sect
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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 b(x) of F [x]. The Euclidean algorithm is an algorithm