Editing Hamming distance (section)

== Error detection and error correction ==
The '''minimum Hamming distance''' or '''minimum distance''' (usually denoted by ''d<sub>min</sub>'') is used to define some essential notions in [[coding theory]], such as [[Error detection and correction|error detecting and error correcting codes]]. In particular, a [[code (coding theory)|code]] ''C'' is said to be ''k'' error detecting if, and only if, the minimum Hamming distance between any two of its codewords is at least ''k''+1.<ref name="Robinson2003">{{cite book |author-first=Derek J. S. |author-last=Robinson |title=An Introduction to Abstract Algebra |date=2003 |publisher=[[Walter de Gruyter]] |isbn=978-3-11-019816-4 |pages=255–257}}</ref>

For example, consider a code consisting of two codewords "000" and "111". The Hamming distance between these two words is 3, and therefore it is ''k''=2 error detecting. This means that if one bit is flipped or two bits are flipped, the error can be detected. If three bits are flipped, then "000" becomes "111" and the error cannot be detected.

A code ''C'' is said to be ''k-error correcting'' if, for every word ''w'' in the underlying Hamming space ''H'', there exists at most one codeword ''c'' (from ''C'') such that the Hamming distance between ''w'' and ''c'' is at most ''k''. In other words, a code is ''k''-errors correcting if the minimum Hamming distance between any two of its codewords is at least 2''k''+1. This is also understood geometrically as any [[Ball (mathematics)#In general metric spaces|closed balls]] of radius ''k'' centered on distinct codewords being disjoint.<ref name="Robinson2003" /> These balls are also called ''[[Hamming sphere]]s'' in this context.<ref name="cc17" />

For example, consider the same 3-bit code consisting of the two codewords "000" and "111". The Hamming space consists of 8 words 000, 001, 010, 011, 100, 101, 110 and 111. The codeword "000" and the single bit error words "001","010","100" are all less than or equal to the Hamming distance of 1 to "000". Likewise, codeword "111" and its single bit error words "110","101" and "011" are all within 1 Hamming distance of the original "111". In this code, a single bit error is always within 1 Hamming distance of the original codes, and the code can be ''1-error correcting'', that is ''k=1''. Since the Hamming distance between "000" and "111" is 3, and those comprise the entire set of codewords in the code, the minimum Hamming distance is 3, which satisfies ''2k+1 = 3''.

Thus a code with minimum Hamming distance ''d'' between its codewords can detect at most ''d''-1 errors and can correct ⌊(''d''-1)/2⌋ errors.<ref name="Robinson2003" /> The latter number is also called the ''[[Sphere packing#Other spaces|packing radius]]'' or the ''error-correcting capability'' of the code.<ref name="cc17">{{citation |title=Covering Codes |volume=54 |series=North-Holland Mathematical Library |author-first1=G. |author-last1=Cohen|author1-link=Gérard Denis Cohen |author-first2=I. |author-last2=Honkala |author-first3=S. |author-last3=Litsyn |author-first4=A. |author-last4=Lobstein |publisher=[[Elsevier]] |date=1997 |isbn=978-0-08-053007-9 |pages=16–17}}</ref>