Linear Hamming Code
A linear Hamming code is a type of error-correcting code that is used to detect and correct errors in data transmission. It is a systematic code, meaning that the original message bits are included in the encoded code words. The encoder mapping in a linear Hamming code is linear, which means that it can be represented by a matrix multiplication.

Given Information
In this problem, we are given that a 4-bit message is mapped to a 7-bit code word using a linear Hamming code. We are also given two examples of message to code word mappings:
- The message 0001 is mapped to the code word 0000111.
- The message 0011 is mapped to the code word 1100110.

Finding the Encoder Mapping
To find the encoder mapping for the given linear Hamming code, we need to find the matrix that represents the mapping. We can do this by using the given examples.

Example 1
The message 0001 is mapped to the code word 0000111.
- The first four bits of the code word represent the original message, so the first four bits of the code word are the same as the message: 0000.
- The last three bits of the code word are the parity bits, which are calculated based on the message bits.
- To calculate the parity bits, we can set up a system of equations using the parity-check matrix of the Hamming code. However, in this case, we can see that the last three bits of the code word are all 1s, which means that the parity bits are all 1s. Therefore, we can conclude that the parity bits for this message are 111.
- Combining the message bits and the parity bits, we get the code word 0000111.

Example 2
The message 0011 is mapped to the code word 1100110.
- Again, the first four bits of the code word are the same as the message: 0011.
- To find the parity bits, we can use the same approach as in Example 1. In this case, we can see that the last three bits of the code word are 110, which means that the parity bits are 110.
- Combining the message bits and the parity bits, we get the code word 1100110.

Using the Encoder Mapping
Now that we have two examples of message to code word mappings, we can try to find the encoder mapping for the linear Hamming code. We can do this by comparing the corresponding bits of the message and the code word for each example.

Comparing the first bit of the message and the code word for each example:
- In Example 1, the first bit of the message is 0 and the first bit of the code word is also 0.
- In Example 2, the first bit of the message is 0 and the first bit of the code word is also 1.

Comparing the second bit of the message and the code word for each example:
- In Example 1, the second bit of the message is 0 and the second bit of the code word is also 0.
- In Example 2, the second bit of the message is 0 and the second bit of the code word is also