Hamming weight [SEP] named after
population count, popcount, sideways sum, or bit summation.
History and usage.
The Hamming weight is named after Richard Hamming although he did not originate the notion. The Hamming weight of binary numbers was already used in 1899 by James W. L. Glaisher to give a formula for the number of odd binomial coefficients in a single row of Pascal's triangle. Irving S. Reed introduced a concept, equivalent to Hamming weight in the binary case, in 1954.
Hamming weight is used in several disciplines including information theory
, and formula_9. The arithmetic distance of a word is upper bounded by its hamming weight since any integer can be represented by its standard polynomial form of formula_10 where the formula_11 are the digits in the integer. Removing all the terms where formula_12 will simulate a formula_13 equal to its hamming weight. The arithmetic weight will usually be less than the hamming weight since the formula_14 are allowed to be negative. For example, the integer formula_15 which is formula_16 in binary has a hamming weight of formula_17. This is a quick upper bound