DISCRETE MATHEMATICS
W W L CHEN
c
W W L Chen, 1991, 2008.
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Chapter 10
INTRODUCTION TO CODING THEORY
10.1. Introduction
The purpose of this chapter and the next two is to give an introduction to algebraic coding theory,
which was inspired by the work of Golay and Hamming.
The study will involve elements of algebra,
probability and combinatorics.
Consider the transmission of a message in the form of a string of 0’s
and 1’s. There may be interference (“noise”), and a different message may be received, so we need to
address the problem of accuracy. On the other hand, certain information may be extremely sensitive,
so we need to address the problem of security.
We shall be concerned with the problem of accuracy in this chapter and the next. In Chapter 12, we
shall discuss a simple version of a security code. Here we begin by looking at an example.
Example 10.1.1.
Suppose that we send the string
w
= 0101100. Then we can identify this string with
the element
w
= (0
,
1
,
0
,
1
,
1
,
0
,
0) of the cartesian product
Z
7
2
=
Z
2
×
. . .
×
Z
2

{z
}
7
.
Suppose now that the message received is the string
v
= 0111101. We can now identify this string with
the element
v
= (0
,
1
,
1
,
1
,
1
,
0
,
1)
∈
Z
7
2
. Also, we can think of the “error” as
e
= (0
,
0
,
1
,
0
,
0
,
0
,
1)
∈
Z
7
2
,
where an entry 1 will indicate an error in transmission (so we know that the 3rd and 7th entries have
been incorrectly received while all other entries have been correctly received). Note that if we interpret
w
,
v
and
e
as elements of the group
Z
7
2
with coordinatewise addition modulo 2, then we have
w
+
v
=
e
,
w
+
e
=
v
,
v
+
e
=
w
.
Chapter 10 : Introduction to Coding Theory
page 1 of 5
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Discrete Mathematics
c
W W L Chen, 1991, 2008
Suppose now that for each digit of
w
, there is a probability
p
of incorrect transmission.
Suppose we
also assume that the transmission of any signal does not in any way depend on the transmission of prior
signals. Then the probability of having the error
e
= (0
,
0
,
1
,
0
,
0
,
0
,
1) is
(1

p
)
2
p
(1

p
)
3
p
=
p
2
(1

p
)
5
.
We now formalize the above.
Suppose that we send the string
w
=
w
1
. . . w
n
∈ {
0
,
1
}
n
. We identify this string with the element
w
= (
w
1
, . . . , w
n
) of the cartesian product
Z
n
2
=
Z
2
×
. . .
×
Z
2

{z
}
n
.
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 Spring '08
 RAMASWAMY
 Math, Coding theory, Hamming Code, Zn, Hamming distance, W W L Chen

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