UNIVERSITY OF CALIFORNIA, SAN DIEGO
Electrical & Computer Engineering Department
ECE 259B - Winter Quarter 2011
Probabilistic Coding
Solutions to Problem Set #6
1.
(a) As discussed in class, there are no stopping sets of weight less than 3 in the Tanner graph
G
corresponding to the “Venn diagram” parity check matrix
H
:
H
=
⎛
⎜
⎝
1110100
1011010
1101001
⎞
⎟
⎠
There are 7 stopping sets of size 3 corresponding to the weight-3 codewords with support
sets:
{
4
,
6
,
7
}{
3
,
5
,
6
2
,
5
,
7
2
,
3
,
4
1
,
4
,
5
1
,
3
,
7
1
,
2
,
6
}
.
There are 3 other weight-3 stopping sets:
{
1
,
2
,
3
1
,
2
,
4
1
,
3
,
4
}
.
We denote this set of 10 weight-3 stopping sets by
S
.
(b) If we append to
H
the redundant parity-check equation obtained by adding the 3 rows
of the parity-check matrix
H
, the resulting parity-check matrix
H
1
is:
H
1
=
⎛
⎜
⎜
⎜
⎝
1000111
⎞
⎟
⎟
⎟
⎠
The corresponding Tanner graph
G
1
consists of
G
with an additional check node and its
connecting edges: