Question

Find the Hamming distance between

(a) $0010101 and 1010101$

(b)$110010101 and 100110010$

(c)$111111 and 000011$

(d)$00001000 and 10001000$

Short answer

1. The humming distance,$d\left(0010101, 1010101\right)=1$
2. The humming distance,$d\left(110010101 , 100110010\right)=5$
3. The humming distance,$d\left(111111, 000011\right)=4$
4. The humming distance,$d\left(00001000, 10001000\right)=1$

Step-by-step solution

Definition of humming distance

Let, $u,v\in B\left(n\right)$.The Hamming distance between u and v , $d\left(u,v\right)$, is

the number of coordinates in which u andvdiffer.

(a)Step 2: Finding the humming distance

Given in the question

$u=0010101$

$v=1010101$

So, the number of coordinates in which u and v differ is 1.

Therefore, according to the definition of humming distance, the humming distance $d\left(0010101, 1010101\right)=1$

(b)Step 3: Finding the humming distance 

Given in the question

$u=110010101$

$v=100110010$

So, the number of coordinates in which u and v differ is 5.

Therefore, according to the definition of humming distance, the humming distance $d\left(110010101 , 100110010\right)=5$

(c)Step 4: Finding the humming distance

Given in the question

$u=111111$

$v=000011$

So, the number of coordinates in which u and v differ is 4.

Therefore, according to the definition of humming distance, the humming distance

$d\left(111111, 000011\right)=4$

(d)Step 5:Finding the humming distance 

Given in the question

$u=00001000$

$v=10001000$

So, the number of coordinates in which u and v differ is 1.

Therefore, according to the definition of humming distance, the humming distance $d\left(00001000, 10001000\right)=1$