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enter image description here My question is what does [this construction gives a Hamming Distance a distance of 3] means? Why the Hamming Distance of ASCII letter A is 3? How do you determine the Hamming Distance is 3?

This is an excerpt from Computer Network by Andrew S. Tanenbaum page 206.

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  • $\begingroup$ The Hamming code has Hamming distance 3. A particular letter doesn't have Hamming distance, since distance is a two-argument function: it gives the distance (in Hamming metric) between two binary vectors. $\endgroup$
    – Yuval Filmus
    May 21 '20 at 13:07
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A code is a collection of binary vectors of some length $n$, known as codewords. The Hamming distance between two codewords $x,y$ is the number of positions $i$ such that $x_i \neq y_i$. The minimum distance of a code is the minimum Hamming distance between two different codewords.

For example, the Hamming(7,4) code consists of 16 codewords of length 7:

0000000
1110000
1001100
0111100
0101010
1011010
1100110
0010110
1101001
0011001
0100101
1010101
1000011
0110011
0001111
1111111

You can check that any two codewords differ in either 3 or 4 positions. For example, 1010101 and 0100101 differ in the first 3 positions. Therefore the minimum distance of the code is 3.

There is absolutely no meaning for the Hamming distance of a single codeword. Hamming distance is a property of pairs of codewords.

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Tanenbaum is not saying that ASCII character "A" has a particular Hamming distance. Computerphile has a video that may clear things up.

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The Hamming distance between two codes is the number of times there is a different bit between them, for example between 1011 and 1111 a Hamming distance of 1.

The example that you're seeing is code correction. When you send anything through a cable there could be some errors (change of bits 1 to 0 or 0 to 1). So there are some common algorithms that minimise these errors within the domain of Information Theory. In this example they show the binary code of the letter A before and after being received by destination, a Hamming distance of 3 was found means that 3 bits have changed.

I previously implemented some related functions and more details are explained on my GitHub. You can check my git repository if you're interested.

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