EE 421 – Fall 2010
Homework 2  Solutions
Problem 1
Consider the binary code composed by the 4 codewords:
C={c1=000000, c2=100100, c3=010010, c4=001001}
•
Is this code linear?
No, the code is not linear. For instance, c2+c3= 100100+010010=110110 does not
belong to the code
•
What is its minimum distance?
d(c1,c2)=2, d(c1,c3)=2, d(c1,c4)=2, d(c2,c3)=4, d(c2,c4)=4, d(c3,c4)=4
Æ
dmin=2
Problem 2
Find the lower bound on required minimum distance for the following codes:
•
A singleerror correcting binary code : t=1
Æ
dmin=3
•
A tripleerror correcting binary code:
t=3
Æ
dmin=7
•
A sixerror detecting binary code:
e=6
Æ
dmin=7
since
⎥
⎦
⎥
⎢
⎣
⎢
−
=
2
1
min
d
t
and
1
min
−
=
d
e
Problem 3
Find the length n, the dimension k, the minimum distance, the generating (encoding)
matrix and a possible base for the linear code defined by the following parity check
matrix:
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1
1
0
1
0
0
0
1
1
1
0
1
0
0
1
1
1
0
0
1
0
1
0
1
0
0
0
1
H
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 Spring '11
 Mondin
 Coding theory, Hamming Code, Minimum distance

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