Linear Hamming Code
The linear Hamming code is a type of error-correcting code that is widely used in telecommunications and computer memory systems. It is designed to detect and correct single-bit errors in transmitted data.

Mapping of 4-bit Messages to 7-bit Codewords
In the given problem, a linear Hamming code is used to map 4-bit messages to 7-bit codewords. The encoder mapping is linear, which means that there is a specific mathematical relationship between the input message and the output codeword.

Given Information
We are given two mappings:
1. The message 0001 is mapped to the codeword 0000111.
2. The message 0011 is mapped to the codeword 100110.

Determining the Mapping for Message 0010
To determine the mapping for the message 0010, we can analyze the given information and look for any patterns or relationships between the messages and codewords.

Observation 1:
Comparing the mappings of the messages 0001 and 0011, we can see that the first four bits of the codeword remain the same (0000) for both messages. This indicates that the first four bits of the codeword are likely the parity bits used for error detection and correction.

Observation 2:
Comparing the mappings of the messages 0001 and 0011 again, we notice that the last three bits of the codeword change (111 and 110, respectively). This suggests that the last three bits of the codeword are related to the specific message being encoded.

Determining the Mapping
To determine the mapping for the message 0010, we can start by assuming that the first four bits of the codeword will remain the same (0000). Based on observation 2, we can deduce that the last three bits of the codeword should be different from the mappings of the messages 0001 and 0011.

Since the last three bits of the codeword for the message 0011 are 110, and the last three bits of the codeword for the message 0001 are 111, the last three bits of the codeword for the message 0010 should be different from both 110 and 111.

By analyzing the given options, we find that option D (1100001) satisfies this condition. Therefore, the message 0010 is mapped to the codeword 1100001.

Conclusion
In conclusion, the linear Hamming code is used to map 4-bit messages to 7-bit codewords. We can determine the mapping for a specific message by analyzing the given mappings and looking for patterns or relationships between the messages and codewords. In this case, the message 0010 is mapped to the codeword 1100001.