University of Illinois
Spring 2011
ECE 361:
Problem Set 5: Solutions
Linear Binary Coding; Gaussian Channel Capacity
1.
[Distances and Weights]
(a) For bits
a,b
∈ {
0
,
1
}
,
a
⊕
b
=
a
+
b

2
ab
. Since the Hamming weight of a binary vector is the
arithmetic sum of its coordinates, the result follows.
(b) From the result of part (a), we have that if w
H
(
x
⊕
y
) is an even number, then w
H
(
x
) + w
H
(
y
)
is also an even number. Therefore, it must be that both w
H
(
x
) and w
H
(
y
) are even numbers, or
both are odd numbers. On the other hand, if w
H
(
x
⊕
y
) is an odd number, then w
H
(
x
) + w
H
(
y
)
is also an odd number. Therefore, it must be that one of w
H
(
x
) and w
H
(
y
) is an even number
and the other is an odd number.
2.
[Odds and Evens]
(a) Let
C
0
denote the set of all codewords that have a 0 in the
i
th coordinate and
C
1
denote the set
of all codewords that have a 1 in the
i
th coordinate. Note that
C
0
∩C
1
=
∅
and
C
0
∪C
1
=
C
, and
that
0
∈ C
0
.
If
C
1
=
∅
, then
C
0
=
C
and we are done:
all
the codewords have a 0 in the
i
th coordinate.
Otherwise,
C
1
6
=
∅
, and so there is at least one codeword
y
∈ C
1
. But then, for each
x
∈ C
0
, the
codeword
x
⊕
y
∈ C
1
since
x
⊕
y
obviously has a 1 in the
i
th coordinate. In particular, note that
0
⊕
y
=
y
∈ C
1
is a codeword of this type. Since each
x
∈ C
0
corresponds to a
x
⊕
y
∈ C
1
(and
there may be other codewords in
C
1
not obtained by this procedure), we conclude that
C
0
 ≤ C
1

.
Conversely, for each
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 Spring '09
 Elementary arithmetic, Even and odd functions, Parity, Evenness of zero, Codewords

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