University of California San Diego
Fall 2008
ECE 259A:
Midterm Exam
Instructions:
There are four problems, weighted as shown below. The exam is openbook: you may
use any auxiliary material that you like.
Good luck!
Problem 1.
(25 points)
Suppose that
M
2
binary vectors of length
n
are all at Hamming distance
d
from each other.
a.
Prove that either all the vectors have even weight, or they all have odd weight.
b.
Show that
d
must be even.
c.
If
M
3
, show that there exists a unique vector at distance
d
2
from the three vectors.
Problem 2.
(35 points)
Suppose that we wish to communicate over a binary channel that either transmits the codeword at its
input as is, or inverts the ±rst
i
bits in the codeword for some
i
1, 2, .
. . ,
n
. Thus the set of all possible
errorpatterns in the channel is
n
1
i
0
n
i
:
i
0, 1, .
. .
n
, where
denotes concatenation.
We wish to construct a binary linear code
of length
n
2
m
1
that corrects all the possible error
patterns: for any
c
and any
e
2
m
1
the codeword
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 Fall '08
 Prof.AlexanderVardy
 Coding theory, Hamming Code, 2m, binary linear code, University of California San Diego, 1

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