University of Illinois
Spring 2011
ECE 361:
Second MidSemester Exam
Tuesday April 12, 2010, 9:30 a.m. – 10:50 a.m.
1. Let
x
,
y
, and
z
denote binary vectors of length
n
. The
Hamming distance
d
H
(
x
,
y
) between
x
and
y
is the number of coordinates in which
x
and
y
differ. The
Hamming weight
w
H
(
x
)
of
x
is the number of nonzero entries in
x
, that is, w
H
(
x
) =
d
H
(
x
,
0
) where
0
is the allzeroes
vector. Equivalently, w
H
(
x
) =
∑
i
x
i
where the sum is the ordinary arithmetic sum of 0’s and
1’s, not an XOR summation.
(a) Suppose that
each
of
x
,
y
, and
z
is at the
same
Hamming distance
d >
0 from the
others
.
Mark the box for each true statement among the following.
2
It is possible to find
x
,
y
, and
z
such that w
H
(
x
), w
H
(
y
), and w
H
(
z
) all are
odd
numbers.
2
It is possible to find
x
,
y
, and
z
such that w
H
(
x
), w
H
(
y
), and w
H
(
z
) all are
even
numbers.
2
It is possible to find
x
,
y
, and
z
such that
exactly two
of w
H
(
x
), w
H
(
y
), and w
H
(
z
)
are
even
numbers and the third one is an odd number.
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