Hamming Distance

The Hamming distance between two words of the same length, $ n $, is the number of positions at which the corresponding symbols differ. It measures the dissimilarity between two words.

Formally, for two words $ x = (x_1, \dots, x_n) $ and $ y = (y_1, \dots, y_n) $, the Hamming distance is: $$ \operatorname{Hamming}(x,y) = \sum_{i=1}^{n} \mathbf{1}_{x_i \neq y_i} $$

To calculate the Hamming distance between two binary numbers of the same length: compare the bits position by position and count the number of positions where the bits differ.

The generalized Hamming distance can be applied to any string of characters. It counts the number of positions where symbols differ.

The Hamming distance measures how different two words of the same length are. It is used to:

— compare two strings or sequences of symbols

— quantify the error between expected and observed data

— measure the similarity between two discrete objects

The smaller the distance, the more similar the two words are. If the distance is zero, the two words are identical.

Several codes are based on the Hamming distance, such as the Hamming (7,4) code: 4 data bits, 3 parity bits, or the extended Hamming (8,4) code: 4 data bits, 4 parity bits.