Editing Hamming distance (section)

== History and applications ==

The Hamming distance is named after [[Richard Hamming]], who introduced the concept in his fundamental paper on [[Hamming code]]s, ''Error detecting and error correcting codes'', in 1950.<ref>{{Cite journal|last=Hamming|first=R. W.|date=April 1950|title=Error detecting and error correcting codes|journal=The Bell System Technical Journal|volume=29|issue=2|pages=147–160|doi=10.1002/j.1538-7305.1950.tb00463.x|s2cid=61141773 |issn=0005-8580|url=https://calhoun.nps.edu/bitstream/10945/46756/1/Hamming_1982.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://calhoun.nps.edu/bitstream/10945/46756/1/Hamming_1982.pdf |doi-access=free |hdl=10945/46756 |hdl-access=free |archive-date=2022-10-09 |url-status=live}}</ref> Hamming weight analysis of bits is used in several disciplines including [[information theory]], [[coding theory]], and [[cryptography]].<ref>{{Cite book |last1=Jarrous |first1=Ayman |last2=Pinkas |first2=Benny |title=Applied Cryptography and Network Security |chapter=Secure Hamming Distance Based Computation and Its Applications |series=Lecture Notes in Computer Science |date=2009 |volume=5536 |editor-last=Abdalla |editor-first=Michel |editor2-last=Pointcheval |editor2-first=David |editor3-last=Fouque |editor3-first=Pierre-Alain |editor4-last=Vergnaud |editor4-first=Damien |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=107–124 |doi=10.1007/978-3-642-01957-9_7 |isbn=978-3-642-01957-9|doi-access=free }}</ref>

It is used in [[telecommunication]] to count the number of flipped bits in a fixed-length binary word as an estimate of error, and therefore is sometimes called the '''signal distance'''.<ref name="Ayala2012">{{cite book |author-first=Jose |author-last=Ayala |title=Integrated Circuit and System Design |date=2012 |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-3-642-36156-2 |page=62}}</ref> For ''q''-ary strings over an [[alphabet]] of size ''q''&nbsp;≥&nbsp;2 the Hamming distance is applied in case of the [[Binary symmetric channel|q-ary symmetric channel]], while the [[Lee distance]] is used for [[phase-shift keying]] or more generally channels susceptible to [[synchronization error]]s because the Lee distance accounts for errors of ±1.<ref name="Roth2006">{{cite book |author-first=Ron |author-last=Roth |title=Introduction to Coding Theory |date=2006 |publisher=[[Cambridge University Press]] |isbn=978-0-521-84504-5 |page=298}}</ref> If <math>q = 2</math> or <math>q = 3</math> both distances coincide because any pair of elements from <math display="inline">\mathbb{Z}/2\mathbb{Z}</math> or <math display="inline">\mathbb{Z}/3\mathbb{Z}</math> differ by 1, but the distances are different for larger <math>q</math>.

The Hamming distance is also used in [[systematics]] as a measure of genetic distance.<ref>{{Cite journal|last1=Pilcher|first1=Christopher D.|last2=Wong|first2=Joseph K.|last3=Pillai|first3=Satish K.|date=2008-03-18|title=Inferring HIV Transmission Dynamics from Phylogenetic Sequence Relationships|journal=PLOS Medicine|language=en|volume=5|issue=3|pages=e69|doi=10.1371/journal.pmed.0050069|pmid=18351799|pmc=2267810|issn=1549-1676 |doi-access=free }}</ref>

However, for comparing strings of different lengths, or strings where not just substitutions but also insertions or deletions have to be expected, a more sophisticated metric like the [[Levenshtein distance]] is more appropriate.