SySc 645 Information Theory:
Final Exam. Solutions
1. (30 points) A Hamming code with block length 7 can be characterized by
the matrix
H
=
1
0
1
0
1
0
1
0
1
1
0
0
1
1
0
0
0
1
1
1
1
Each legal input codeword
x
satisfies
Hx
= 0
(all arithmetic is understood to be mod 2).
The 7 bits received can be
represented by
y
=
x
+
e
where
e
is the error pattern. Assume that the
bits are transmitted over a memoryless binary symmetric channel with
the probability of error in each bit of
which is much less than
1
2
.
(a) How many codewords (legal input
x
’s) are there?
Answer:
16
(b) If
y
= 1110010 what is ˆ
x
(
y
), the best guess for
x
?
Answer:
1110000
(c) Assuming that you attempt to correct errors in received words, find
P
E
, the probability that you will make an error in guessing the trans
mitted codeword:
i. First write out an exact algebraic expression.
Answer:
P
E
= 1

(1

)
7

7 (1

)
6
ii. Give an approximate numerical value for the expression assuming
that
= 10

9
.
Answer:
P
E
≈
(
7
2
)
2
= 21
×
10

18
2. (20 points) Given the transition probabilities
P
Y

X
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 '09
 Information Theory, Coding theory, Hamming Code, 7 bits, 1 bits, legal input codeword

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