Question

A digital transmission system uses a (7,4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code \( (m_i ; c_i) \), where \( c_i \) is the codeword corresponding to the \( i^{th} \) message \( m_i \), are known to be \[ (1 1 0 0 ; 0 1 0 1 1 0 0), \quad (0 0 1 1 1 1 0 ; 0 1 1 1 0 ; 1 0 0 0 1 1 0), \] then which of the following is a valid codeword in this code?

• A. 1 1 0 1 0 0 1
• B. 1 0 1 1 0 1 0
• C. 0 0 0 1 0 1 1
• D. 0 1 1 0 1 0 0

Hint:

In systematic linear Hamming codes, ensure that the structure of the codeword is maintained with message bits followed by parity check bits. The parity check should satisfy the necessary conditions for error detection and correction.

Answer 1

The Correct Option is C

In a Hamming code, we have the following systematic encoding procedure:
- The first \( k = 4 \) bits represent the message, while the remaining \( n - k = 3 \) bits are the parity check bits.
- The Hamming code ensures that the codeword contains enough redundancy to detect and correct single-bit errors.
Given the message-codeword pairs: - \( (1 1 0 0 ; 0 1 0 1 1 0 0) \)
- \( (0 0 1 1 1 1 0 ; 0 1 1 1 0 ; 1 0 0 0 1 1 0) \)
Step 1: Identify the structure of the codeword.
The valid codewords should be of the form \( (m_1, m_2, m_3, m_4 ; c_1, c_2, c_3) \), meaning the 4 message bits \( m_1, m_2, m_3, m_4 \) are followed by 3 parity check bits. Step 2: Check the validity of the options.
- Option (A): \( 1 1 0 1 0 0 1 \) does not follow the structure as it violates the parity checking criteria. - Option (B): \( 1 0 1 1 0 1 0 \) does not follow the systematic Hamming code format.
- Option (C): \( 0 0 0 1 0 1 1 \) is a valid codeword based on the Hamming code structure and satisfies the parity check equations.
- Option (D): \( 0 1 1 0 1 0 0 \) does not follow the structure as it violates the parity checking criteria.
Thus, the correct answer is option (C). Final Answer: 0 0 0 1 0 1 1