## Hamming Distance February 21, 2006

The *Hamming Distance* between two words is the number of locations (or indeces, or coordinate) by which they differ; If a code *C* contains two code words which are displaced by Hamming distance δ and there is no pair of distinct code words with distance less than δ, then δ is called the minimum distance of *C*, written *d*(*C*) = δ

The weight of a code *x*, *w*(*x*) is the number of position not occupied by zero.

**[Theorem]**

For any two binary words *x* and *y*,

*d*(*x*, *y*) = *w*(*x* + *y*) = *w*(*x*) + *w*(*y*) – *2w*(*x*⋅*y*)

where “⋅” is the bitwise product modulo 2.

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