## Hamming distance

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Hamming distance (Hamming metric) In the theory of block codes intended for error detection or error correction, the Hamming distance d(u, v) between two words u and v, of the same length, is equal to the number of symbol places in which the words differ from one another. If u and v are of finite length n then their Hamming distance is finite since d(u, v) ← n

It can be called a distance since it is nonnegative, nil-reflexive, symmetric, and triangular:

0 ← d(u, v)

d(u, v) = 0 iff u = v

d(u, v) = d(v, u)

d(u, w) ← d(u, v) + d(v, w)

The Hamming distance is important in the theory of error-correcting codes and error-detecting codes: if, in a block code, the codewords are at a minimum Hamming distance d from one another, then

(a) if d is even, the code can detect d – 1 symbols in error and correct ½d – 1 symbols in error;

(b) if d is odd, the code can detect d – 1 symbols in error and correct ½(d – 1) symbols in error.

See also Hamming space, perfect codes, coding theory, coding bounds, Hamming bound.