M 375 – Homework 10 Page 1
M 375 – Homework 10
4.6.1
Parity check matrix H:
=
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
H
The columns of the parity check matrix are the binary representation of the number 1…15.
(a) syndrome = 0001 first digit of codeword is wrong:
correct codeword (000 000 000 000 000).
(b) syndrome = 0000 no error: (111 111 111 111 111).
4.7.1
There are 8 possible binary codes of length 3.
Those are
{
}
)
111
(
),
110
(
),
101
(
),
100
(
),
011
(
),
010
(
),
001
(
),
000
(
. But only 4 of them are linearly
independent: a) (000): 0
b) (001): 1
c) (011):
1
+
x
d) (111):
1
2
+
+
x
x
All other codes can be produced by cyclic permutation of a) – d):
(010) and (100): permute b) once and twice, respectively
(101) and (110): permute c) once and twice, respectively.
4.7.2
Known from the lecture:
1
)
(
)
(
7
=
x
x
h
x
g
and
1
)
(
3
+
+
=
x
x
x
g
.
1
)
1
)(
1
(
2
4
2
3
5
3
4
5
7
2
4
3
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+
+
+
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
.
In
]
[
F
7
2
x
this polynomial is reduced to:
1
7
x
.
