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| In [[coding theory]], a '''constant-weight code''', also called an '''m of n code''', is an [[error detection and correction]] code where all codewords share the same [[Hamming weight]]. The theory is closely connected to that of [[Combinatorial design|designs]] (such as [[block design|''t''-design]]s and [[Steiner system]]s). Most of the work on this very vital field of [[discrete mathematics]] is concerned with ''binary'' constant-weight codes.
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| Binary constant-weight codes have several applications, including [[Frequency-hopping spread spectrum|frequency hopping]] in [[Global System for Mobile Communications|GSM]] networks.<ref name="smith">D. H. Smith, L. A. Hughes and S. Perkins (2006). "[http://www.combinatorics.org/Volume_13/Abstracts/v13i1a2.html A New Table of Constant Weight Codes of Length Greater than 28]". ''The Electronic Journal of Combinatorics'' '''13'''.</ref>
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| Most [[barcode]]s use a binary constant-weight code to simplify automatically setting the threshold.
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| Most [[line code]]s use either a constant-weight code, or a nearly-constant-weight [[paired disparity code]].
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| In addition to use as error correction codes, the large space between code words can also be used in the design of [[asynchronous circuit]]s such as [[delay insensitive circuit]]s.
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| Constant-weight codes, like [[Berger code]]s, can detect all unidirectional errors.
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| == ''A''(''n'',''d'',''w'') ==
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| The central problem regarding constant-weight codes is the following: what is the maximum number of codewords in a binary constant-weight code with length <math>n</math>, [[Hamming distance]] <math>d</math>, and weight <math>w</math>? This number is called <math>A(n,d,w)</math>. | |
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| Apart from some trivial observations, it is generally impossible to compute these numbers in a straightforward way. Upper bounds are given by several important theorems such as the ''first and second Johnson bounds'',<ref>See pp. 526–527 of F. J. MacWilliams and N. J. A. Sloane (1979). ''The Theory of Error-Correcting Codes''. Amsterdam: North-Holland.</ref> and better upper bounds can sometimes be found in other ways. Lower bounds are most often found by exhibiting specific codes, either with use of a variety of methods from discrete mathematics, or through heavy computer searching. A large table of such record-breaking codes was published in 1990,<ref name="brouwer">A. E. Brouwer, James B. Shearer, N. J. A. Sloane and Warren D. Smith (1990). "A New Table of Constant Weight Codes". ''IEEE Transactions of Information Theory'' '''36'''.</ref> and an extension to longer codes (but only for those values of <math>d</math> and <math>w</math> which are relevant for the GSM application) was published in 2006.<ref name="smith"/>
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| == 1 of N codes ==
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| {{main|one-hot}}
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| A special case of constant weight codes are the one-of-N codes, that encode <math>log_2 N</math> bits in a code-word of <math>N</math> bits. The one-of-two code uses the code words 01 and 10 to encode the bits '0' and '1'. a one-of-four code can use the words 0001, 0010, 0100, 1000 in order to encode two bits 00, 01, 10, and 11. An example is [[dual rail encoding]], and chain link <ref>{{cite news
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| | title=System-on-Chip Design using Self-timed Networks-on-Chip
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| | url=http://www.design-reuse.com/articles/14561/system-on-chip-design-using-self-timed-networks-on-chip.html
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| | authors=W.J.Bainbridge, A.Bardsley, R.W.McGuffin
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| }}</ref> used in delay insensitive circuits.
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| == m of n codes==
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| An '''''m'' of ''n'' code''' is a separable [[error detection]] code with a code word length of ''n'' bits, where each code word contains exactly ''m'' instances of a "one." A single bit error will cause the code word to have either ''m'' + 1 or ''m'' – 1 "ones". An example ''m''-of-''n'' code is the [[Two-out-of-five code|2 of 5 code]] used by the [[United States Postal Service]].
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| The simplest implementation is to append a string of ones to the original data until it contains ''m'' ones, then append zeros to create a code of length ''n''.
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| Example:
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| {|class="wikitable" style="text-align:center"
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| |+ 3 of 6 code
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| |-
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| ! Original 3 data bits !! Appended bits
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| |-
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| |000 || 111
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| |-
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| |001 || 110
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| |-
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| |010 || 110
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| |-
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| |011 || 100
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| |-
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| |100 || 110
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| |-
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| |101 || 100
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| |-
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| |110 || 100
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| |-
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| |111 || 000
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| |-
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| |}
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| Some of the more notable uses of constant-weight codes include
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| [[biphase mark code]] uses a 1 of 2 code;
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| [[pulse-position modulation]] uses a 1 of n code;
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| [[Code 39]] uses a 3 of 9 code;
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| [[6b/8b encoding]] uses a 4 of 8 code;
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| [[bi-quinary coded decimal]] code uses a 2 of 7 code,
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| etc.
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| == References ==
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| <!--See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref> and </ref> tags and the tag below -->
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| <references/>
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| == External links ==
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| * [http://www.win.tue.nl/~aeb/codes/Andw.html Table of lower bounds on <math>A(n,d,w)</math>] maintained by [[Andries Brouwer]] (update of the [http://www.research.att.com/~njas/codes/Andw/index.html earlier table] by [[Neil Sloane]] and [[E. M. Rains]])
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| * [http://webfiles.portal.chalmers.se/s2/research/kit/bounds/ Table of upper bounds on <math>A(n,d,w)</math>] maintained by [[Erik Agrell]]
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| [[Category:Information theory]]
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| [[Category:Error detection and correction]]
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