A linear Hamming code is used to map 4-bit messages to 7-bit code words. The encoder mapping is linear. If the message 0001 is mapped to the code word 0000111, and the message 0011 is mapped to the code word 1100110, then the message 0010 is mapped to

Hamming Code Encoder Explained

The question describes a situation where a linear Hamming code is used. This type of code maps 4-bit messages to 7-bit codewords. A crucial property mentioned is that the encoder mapping is linear. Understanding what "linear" means in the context of coding theory is key to solving this problem.

In a linear code, if two messages, say ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$, are mapped to codewords ${\mathbf{c}}_1$ and ${\mathbf{c}}_2$ respectively, then the message formed by the bit-wise XOR (exclusive OR) sum of ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$, i.e., ${\mathbf{m}}_1\oplus {\mathbf{m}}_2$, will be mapped to the codeword formed by the bit-wise XOR sum of ${\mathbf{c}}_1$ and ${\mathbf{c}}_2$, i.e., ${\mathbf{c}}_1\oplus {\mathbf{c}}_2$. This property holds true for any combination of messages in a linear code.

Given Message and Codeword Mappings

We are provided with the following two mappings for the linear Hamming code:

- **Message 1 (${\mathbf{m}}_1$):** $0001$ is mapped to **Codeword 1 (${\mathbf{c}}_1$):** $0000111$
- **Message 2 (${\mathbf{m}}_2$):** $0011$ is mapped to **Codeword 2 (${\mathbf{c}}_2$):** $1100110$

Our goal is to find the codeword for the message $0010$. Let's call this target message ${\mathbf{m}}_{target}$ and its unknown codeword ${\mathbf{c}}_{target}$.

Applying Linearity to Find the Codeword

First, let's examine the relationship between the given messages and the target message using the XOR operation (modulo 2 addition):

- Let ${\mathbf{m}}_1=0001$
- Let ${\mathbf{m}}_2=0011$
- Let ${\mathbf{m}}_{target}=0010$

We can observe that if we XOR ${\mathbf{m}}_1$ with ${\mathbf{m}}_{target}$, we get ${\mathbf{m}}_2$:

${\mathbf{m}}_1\oplus {\mathbf{m}}_{target}=0001\oplus 0010=0011$

This means: $0001\oplus 0010=0011$

Since the encoder mapping is linear, the same relationship must hold true for their corresponding codewords. Therefore, if ${\mathbf{c}}_1$ corresponds to ${\mathbf{m}}_1$, ${\mathbf{c}}_{target}$ corresponds to ${\mathbf{m}}_{target}$, and ${\mathbf{c}}_2$ corresponds to ${\mathbf{m}}_2$, then:

${\mathbf{c}}_1\oplus {\mathbf{c}}_{target}={\mathbf{c}}_2$

To find ${\mathbf{c}}_{target}$, we can rearrange the equation by XORing both sides with ${\mathbf{c}}_1$. Remember that $\mathbf{X}\oplus \mathbf{X}=0$:

${\mathbf{c}}_{target}={\mathbf{c}}_1\oplus {\mathbf{c}}_2$

Calculating the Unknown Codeword

Now, let's perform the bit-wise XOR operation on the given codewords:

${\mathbf{c}}_1=0000111$

${\mathbf{c}}_2=1100110$

Calculating ${\mathbf{c}}_1\oplus {\mathbf{c}}_2$:Therefore, the codeword for the message $0010$ is $1100001$.

This result aligns with one of the provided options.