A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message code-word pairs in this code (mi;ci ) , where, ci is the code-word corresponding to the i th message mi , are known to be (1100 ; 0101100) , (1110 ; 0011110) , (0110 ; 1000110) , then which of the following is a valid code-word in this code?a)1101001b)1011010c)0001011d)0110100Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Systematic Linear Hamming Code (7,4)

A systematic linear Hamming code (7,4) is a type of error-correcting code that adds three parity bits to a 4-bit message to form a 7-bit codeword. The three parity bits are calculated based on the values of the message bits. The code is designed to detect and correct single-bit errors in transmission.

Given Information

We are given three message code-word pairs in this code, (1100 ; 0101100), (1110 ; 0011110), and (0110 ; 1000110).

Valid Code-Word

We need to find a valid code-word in this code. To do this, we can use the following steps:

1. Calculate the parity bits for each of the message code-word pairs.

For (1100 ; 0101100), the parity bits are calculated as follows:

p1 = m1 + m2 + m4 = 1 + 1 + 0 = 0
p2 = m1 + m3 + m4 = 1 + 0 + 0 = 1
p3 = m2 + m3 + m4 = 1 + 0 + 0 = 1

So the full code-word is 0101100.

For (1110 ; 0011110), the parity bits are calculated as follows:

p1 = m1 + m2 + m4 = 1 + 1 + 0 = 0
p2 = m1 + m3 + m4 = 1 + 1 + 0 = 0
p3 = m2 + m3 + m4 = 1 + 1 + 0 = 0

So the full code-word is 0011110.

For (0110 ; 1000110), the parity bits are calculated as follows:

p1 = m1 + m2 + m4 = 0 + 1 + 0 = 1
p2 = m1 + m3 + m4 = 0 + 0 + 0 = 0
p3 = m2 + m3 + m4 = 1 + 0 + 0 = 1

So the full code-word is 1000110.

2. Compare the parity bits of the known code-words to the parity bits of the possible code-words.

We can use this method to check each of the possible code-words:

For (1101 ; ?), the parity bits are calculated as follows:

p1 = m1 + m2 + m4 = 1 + 1 + 1 = 1
p2 = m1 + m3 + m4 = 1 + 0 + 1 = 0
p3 = m2 + m3 + m4 = 1 + 0 + 1 = 0

So the full code-word would be 0110101, which does not match any of the known code-words.

For (1011 ; ?), the parity bits are calculated as follows:

p1 = m1 + m2 + m4 = 1 + 0 + 1 = 0
p2 = m1 + m3 + m4 = 1 + 1 + 1 = 1
p3 = m2 + m3 + m4 = 0 + 1 +

Answer

Given message and corresponding code-words are

Here, m₁₂ means, m₁m₂ m₃ m₄ = 1100 and its decimal equivalent is 12

Here, m₁₄ means, m₁m₂m₃ m₄ = 1110 and its decimal equivalent is 14

Here, m₆ means, m₁m₂m₃ m₄= 0110 and its decimal equivalent is 6

From four parity generator we have to select any three equation,

P_A =m₁⊕m₂ ⊕ m₃

P_B =m₁⊕m₂ ⊕ m₄

P_C =m₂⊕ m₃ ⊕ m₄

P_D =m₁⊕ m₃ ⊕ m₄

From comparing with all three given code-words i.e., C₁₂ ,C₁₄ and C₆

P₁=P_B,P₂=P_C,P₃=P_A

So, for message m₉ = 1001,C₉= 0111001

Option (A) is wrong.

m₁₀ = 1010,C₁₀= 1101010

Option (B) is wrong.

m₁₁= 1011,C₁₁= 0001011

Option (C) is correct.

m₄= 0100,C₄= 1110100

Option (D) is wrong.

Hence, the correct option is (C).

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