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Homework #7
Solutions
Due: October 26, 2011
1. Let
C
‰
(
F
2
)
9
be the subset
C
=
'
abcabc
a
b
c
j
a; b; c
2
F
2
“
;
where
x
is 1 if
x
is 0 and
x
is 0 if
x
is 1. Then
C
is a binary code of length 9.
(a) List all of the codewords in
C
.
I
Solution.
The set of codewords is obtained by letting
a
,
b
,
c
vary over
F
2
,
that is,
a
,
b
,
c
can be assigned either 0 or 1. Thus,
C
=
f
000000111
;
001001110
;
010010101
;
011011100
;
100100011
;
101101010
;
110110001
;
111111000
g
:
J
(b) Show that
C
is not a linear code.
I
Solution.
A linear code must contain the 0 vector, but 000000000 is not in
C
.
Alternatively, one can observe that 000000111+001001110 = 001001001 which is
not in
C
, so
C
is not closed under addition.
J
(c) How many errors does
C
detect and how many errors does it correct.
I
Solution.
It is necessary to compute the minimum distance
d
(
C
) for the code.
Since the code is not linear, it is necessary to compute the distance between any
two code words. But codewords in this code are determined by the ﬂrst 3 digits,
the second 3 digits repeat the ﬂrst 3, and the third 3 digits are the opposites of
the ﬂrst 3. Thus, for each digit that two codewords diﬁer in the ﬂrst 3 digits,
this diﬁerence is replicated in the second and third blocks of 3 digits. Hence, the
minimum distance must be 3, which corresponds to 1 diﬁerence in the ﬂrst block
of 3 digits. Thus,
d
=
d
(
C
) = 3, so this code detects
d
¡
1 = 2 errors and corrects
b
d
¡
1
2
c
= 1 error.
J
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