The Hamming distance between two binary strings of the same length is the number of positions at which the corresponding bits are different. It's a measure of how many substitutions are needed to change one string into the other.

Calculation of Hamming Distance

To calculate the Hamming distance between two binary strings:

1. Compare each bit of the two strings.
2. Count the number of positions where the bits differ.

Applying Hamming Distance to the Given Problem

Given codewords: $000$, $011$, $101$, $110$.

Let's calculate the Hamming distance between each pair of codewords:

1. Hamming distance between $000$ and $011$:

   • Comparing each bit:
     • $0 \neq 0$
     • $0 \neq 1$
     • $0 \neq 1$
   • Hamming distance: 2 (second and third bits differ).

2. Hamming distance between $000$ and $101$:

   • Comparing each bit:
     • $0 \neq 1$
     • $0 \neq 0$
     • $0 \neq 1$
   • Hamming distance: 2 (first and third bits differ).

3. Hamming distance between $000$ and $110$:

   • Comparing each bit:
     • $0 \neq 1$
     • $0 \neq 1$
     • $0 \neq 0$
   • Hamming distance: 2 (first and second bits differ).

4. Hamming distance between $011$ and $101$:

   • Comparing each bit:
     • $0 \neq 1$
     • $1 \neq 0$
     • $1 \neq 1$
   • Hamming distance: 2 (first and second bits differ).

5. Hamming distance between $011$ and $110$:

   • Comparing each bit:
     • $0 \neq 1$
     • $1 \neq 1$
     • $1 \neq 0$
   • Hamming distance: 2 (first and third bits differ).

6. Hamming distance between $101$ and $110$:

   • Comparing each bit:
     • $1 \neq 1$
     • $0 \neq 1$
     • $1 \neq 0$
   • Hamming distance: 2 (second and third bits differ).

Summary of Hamming Distances

• $000$ and $011$: 2
• $000$ and $101$: 2
• $000$ and $110$: 2
• $011$ and $101$: 2
• $011$ and $110$: 2
• $101$ and $110$: 2

Graph Representation

Since none of the pairs of codewords have a Hamming distance of 1, the graph will consist of 4 isolated vertices with no edges connecting them.

000    011    101    110

Would you like further details or have any questions on this topic?

Questions for Further Exploration:

1. What are Tanner graphs and how are they used in decoding?
2. How do trellis graphs assist in the decoding of convolutional codes?
3. What is the relationship between graph theory and network coding?
4. Can you explain the concept of LDPC codes and their graph representation?
5. How does the Hamming distance impact the construction of graphs in coding theory?

Tip:

When calculating the Hamming distance, ensure each bit position is compared correctly to avoid errors, especially when dealing with longer codewords or more complex codes.