Introduction to Algebraic Coding Theory
Supplementary material for Math 336 Cornell University
Sarah A. Spence
Contents
1
Introduction
1
2
Basics
2
2.1
Important code parameters
. . . . . . . . . . . . . . . . . . . . .
4
2.2
Correcting and detecting errors
. . . . . . . . . . . . . . . . . . .
5
2.3
Spherepacking bound
. . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3
Linear codes
9
3.1
Generator and parity check matrices
. . . . . . . . . . . . . . . .
11
3.2
Coset and syndrome decoding
. . . . . . . . . . . . . . . . . . . .
14
3.3
Hamming codes
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.4
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4
Ideals and cyclic codes
19
4.1
Ideals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.2
Cyclic codes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.3
Group of a code
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.4
Minimal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . .
31
4.5
BCH and ReedSolomon codes
. . . . . . . . . . . . . . . . . . .
33
4.6
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5
Acknowledgements
40
1
Introduction
Imagine that you are using an infrared link to beam messages consisting of 0s
and 1s from your laptop to your friend’s PalmPilot.
Usually, when you send
a 0, your friend’s PalmPilot receives a 0. Occasionally, however,
noise
on the
channel causes your 0 to be received as a 1. Examples of possible causes of noise
include atmospheric disturbances. You would like to find a way to transmit your
messages in such a way that errors are detected and corrected. This is where
errorcontrol codes
come into play.
1
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Errorcontrol codes
are used to detect and correct errors that occur when
data is transmitted across some noisy channel. Compact discs (CDs) use error
control codes so that a CD player can read data from a CD even if it has been
corrupted by noise in the form of imperfections on the CD. When photographs
are transmitted to Earth from deep space, errorcontrol codes are used to guard
against the noise caused by lightning and other atmospheric interruptions.
Errorcontrol codes build redundancy into a message. For example, if your
message is
x
= 0, you might encode
x
as the
codeword
c
= 00000. (We work
more with this example in Chapter 2.) In general, if a message has length
k
,
the encoded message,
i.e.
codeword, will have length
n > k
.
Algebraic coding theory is an area of discrete applied mathematics that is
concerned (in part) with developing errorcontrol codes and encoding/decoding
procedures. Many areas of mathematics are used in coding theory, and we focus
on the interplay between algebra and coding theory. The topics in this packet
were chosen for their importance to developing the major concepts of coding
theory and also for their relevance to a course in abstract algebra. We aimed to
explain coding theory concepts in a way that builds on the algebra learned in
Math 336. We recommend looking at any of the books in the bibliography for
a more detailed treatment of coding theory.
As you read this packet, you will notice questions interspersed with the text.
These questions are meant to be straightforward checks of your understanding.
You should work out each of these problems as you read the packet.
More
homework problems are found at the end of each chapter.
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 Spring '08
 BEHREND
 Algebra, Coding theory, codeword

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