In short, a '''Hamming''''''Code''' of separation '''s''' and length '''n''' is a collection of '''n''' "words" from an "alphabet" chosen such that every pair of distinct words are at least '''s''' apart, according to the HammingDistance.

In slightly more detail ...

Take an alphabet '''A''' and consider the set '''W''' of all the "words" of length '''n''' that can be made from it.

The HammingDistance '''d(,)''' between two "words" is the number of places in which they differ.

A HammingCode of distance '''s''' is a subset of '''S''' of '''W''' such that for any two words chosen from '''S''' have a HammingDistance of at least '''s'''

'''Examples:'''

Let '''A = {0,1}''' and '''n=3''', so '''|W|=8'''. Let '''S''' be '''{000,101,110,011}''', a code obtained by prepending the parity to each of '''{00,01,10,11}'''.  '''S''' has separation 2 and single-bit errors in transmission can be detected.

Let '''A = {0,1}''' and '''n=7''', so '''|W|=128'''. It is possible to find a set S of separation 3 and size 16 in this case. Furthermore, the set '''S''' can be chosen such that the bottom four bits of the code words are the binary representation of the numbers 0 to 15. It's like the parity example, but harder, and bigger.

'''Motivation:'''

If a word is transmitted and some bits are corrupted, we would like to know and, if possible, be able to recover the original. Given a HammingCode of separation 3 we can always recover from a single-bit error. Given a HammingCode of separation 4 we can always recover from a single-bit error, and detect a double-bit error.
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Not a crystal clear explanation, but it can be expanded.