[amp_mcq option1=”10011″ option2=”1111111″ option3=”1111000″ option4=”1100001″ correct=”option1″]
The correct answer is $\boxed{\text{A}}$.
A linear Hamming code is a linear code that can correct single-bit errors. It is a $7$-bit code, which means that it can encode $2^7 = 128$ messages. The encoder mapping is linear, which means that the codeword for a message is the sum of the codewords for the individual bits of the message.
If the message $0001$ is mapped to the codeword $0000111$, and the message $0011$ is mapped to the codeword $1100110$, then the message $0010$ is mapped to the codeword $0010011$. This is because the codeword for $0010$ is the sum of the codewords for $0001$ and $0011$, which is $0000111 + 1100110 = 0010011$.
Here is a more detailed explanation of each option:
- Option A: $0010011$ is the correct answer. This is because the codeword for $0010$ is the sum of the codewords for $0001$ and $0011$, which is $0000111 + 1100110 = 0010011$.
- Option B: $1111111$ is not the correct answer. This is because the codeword for $0010$ cannot be $1111111$, since $1111111$ is not a valid codeword in a $7$-bit Hamming code.
- Option C: $1111000$ is not the correct answer. This is because the codeword for $0010$ cannot be $1111000$, since $1111000$ is not a valid codeword in a $7$-bit Hamming code.
- Option D: $1100001$ is not the correct answer. This is because the codeword for $0010$ cannot be $1100001$, since $1100001$ is not a valid codeword in a $7$-bit Hamming code.