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141
Solution.
Absent.
(ii)
Prove that

B
r
(
u
)
=
r
X
i
=
0
³
n
i
´
(
q
−
1
)
i
.
Solution.
Absent.
(iii)
(
GilbertVarshamov bound
.
)I
f
C
⊆
A
n
is an
(
n
,
M
,
d
)
code,
where

A
=
q
, prove that
q
n
∑
d
−
1
i
=
0
(
n
i
)
(
q
−
1
)
i
≤
M
.
Solution.
Absent.
4.70
(
Hamming bound
)If
C
⊆
A
n
is an
(
n
,
M
,
d
)
code, where

A
=
q
and
d
=
2
t
+
1, prove that
M
≤
q
n
∑
t
i
=
0
(
n
i
)
(
q
−
1
)
i
.
Solution.
Absent.
4.71
Prove that the Hamming
[
2
`
−
1
,
2
`
−
1
−
`
]
codes in Example 4.113 are
perfect codes.
Solution.
Absent.
4.72
Suppose that a code
C
detects up to
s
errors and that it corrects up to
t
errors. Prove that
t
≤
s
.
Solution.
Absent.
4.73
If
C
⊆
F
n
is a linear code and
w
∈
F
n
,de
f
ne
r
=
min
c
∈
C
δ(w,
c
)
.G
ive
an example of a linear code
C
⊆
F
n
, which corrects up to
t
errors, and a
word
w
∈
F
n
with
w/
∈
C
, such that there are distinct codewords
c
,
c
0
∈
C
with
δ(w,
c
)
=
r
=
δ(
x
,
c
0
)
. Conclude that correcting a transmitted word
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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