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 weight3 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 weight3 stopping sets:
{
1
,
2
,
3
1
,
2
,
4
1
,
3
,
4
}
.
We denote this set of 10 weight3 stopping sets by
S
.
(b) If we append to
H
the redundant paritycheck equation obtained by adding the 3 rows
of the paritycheck matrix
H
, the resulting paritycheck 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:
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0
c
1
c
2
c
3
c
4
c
5
c
6
If we denote the set of weight3 stopping sets in
G
1
by
S
1
, then necessarily
S
1
⊂
S
,s
ince
adding redundant checks cannot create new stopping sets. Also
S
1
must contain the 7
stopping sets corresponding to the supports of the 7 weight3 codewords. By inspection,
we see that the new check node has exactly one edge connecting to the other 3 stopping
sets in
S
, namely the edge connecting to variable node 1. Therefore, these sets are not
stopping sets in
G
1
.
2.
(a) The rate
1
5
repetition code
C
has 2 length5 codewords, namely the allzeros and allones
codewords
,[00000]and[11111]
.
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 Spring '11
 Song
 Linear Algebra, li, vmax, Coding theory, Hamming Code, Tanner graph

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