Hamming distance

The Hamming distance between two vectors of equal length is the number of positions at which the corresponding entries differ.

In other words, it measures the minimum number of substitutions required to change one codeword into another.

Formally,
let $F$ be a field, and let $\overrightarrow{x},\overrightarrow{y}\in F$.
The Hamming Distance on $F^n$ is defined by

i.e. the number of positions where $\overrightarrow{x}$ and $\overrightarrow{y}$ disagree.

The Hamming distance is a metric on the Hamming space, since it is

• non-negative,
• symmetric,
• $d_H(\overrightarrow{x},\overrightarrow{y})=0$ if and only if $\overrightarrow{x}=\overrightarrow{y}$,
• satisfies the triangle inequality

Hamming distance for binary codes can be calculated using the binomial coefficient.

See also
minimum Hamming distance
Hamming ball
error detection and correction