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== Algorithm ==
The validity of a digit sequence containing a check digit is defined over a quasigroup. A quasigroup table ready for use can be taken from Damm's dissertation (pages 98, 106, 111).<ref name="dhmd" /> It is useful if each main diagonal entry is {{math|0}},<ref name=fenwick2014 /> because it simplifies the check digit calculation.
=== Validating a number against the included check digit ===
# Set up an interim digit and initialize it to {{math|0}}.
# Process the number digit by digit: Use the number's digit as column index and the interim digit as row index, take the table entry and replace the interim digit with it.
# The number is valid if and only if the resulting interim digit has the value of {{math|0}}.<ref name=fenwick2014 />
=== Calculating the check digit ===
'''Prerequisite:''' The main diagonal entries of the table are {{math|0}}.
#Set up an interim digit and initialize it to {{math|0}}.
#Process the number digit by digit: Use the number's digit as column index and the interim digit as row index, take the table entry and replace the interim digit with it.
#The resulting interim digit gives the check digit and will be appended as trailing digit to the number.<ref name=fenwick2014 />
== Example ==
The following operation table will be used.<ref name="fenwick2014" /> It may be obtained from the totally anti-symmetric quasigroup {{math|''x'' ∗ ''y''}} in Damm's doctoral dissertation page 111<ref name="dhmd" /> by rearranging the rows and changing the entries with the permutation {{math|1=''φ'' = (1 2 9 5 4 8 6 7 3)}} and defining {{math|1=''x'' ⋅ ''y'' = ''φ''<sup>
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The resulting interim digit is '''4'''. This is the calculated check digit. We append it to the number and obtain '''5724'''.
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The resulting interim digit is '''0''', hence the number is '''valid'''.
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<ref name="dhmd">{{cite book |last=Damm |year=2004 |first=H. Michael |title=Total anti-symmetrische Quasigruppen |type=Dr. rer. nat. |publisher=Philipps-Universität Marburg |url=http://archiv.ub.uni-marburg.de/diss/z2004/0516/pdf/dhmd.pdf |id=[http://nbn-resolving.de/urn:nbn:de:hebis:04-z2004-05162 urn:nbn:de:hebis:04-z2004-05162]|language=de}}</ref>
<ref name="damm2003">{{cite journal |last=Damm |year=2003 |first=H. Michael |title=On the Existence of Totally Anti-Symmetric Quasigroups of Order 4''k'' + 2 |journal=Computing |volume=70 |issue=4 |pages=349–357 |issn=0010-485X |doi=10.1007/s00607-003-0017-3 |s2cid=31659430 }}</ref>
<ref name="damm2007">{{cite journal |last=Damm |year=2007 |first=H. Michael |title=Totally anti-symmetric quasigroups for all orders ''n''
<ref name="fenwick2014">{{cite book |last=Fenwick |year=2014 |first=Peter |editor1-first=Peter |editor1-last=Fenwick |title=Introduction to Computer Data Representation |chapter=Checksums and Error Control |doi=10.2174/9781608058822114010013 |pages=[http://ebooks.benthamscience.com/sample/9781608058822/51/ 191–218] |publisher=Bentham Science Publishers |isbn=978-1-60805-883-9 }}</ref>
<ref name="Salomon2005">For the types of common errors and their frequencies, see {{cite book |last=Salomon |year=2005 |first=David |title=Coding for Data and Computer Communications |publisher=Springer Science+Business Media, Inc. |pages=36 |url=https://books.google.com/books?id=Zr9bjEpXKnIC&pg=PA36 |isbn=978-0387-21245-6 }}</ref>
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