Damm algorithm: Difference between revisions

Revision as of 05:02, 11 May 2020

Revision as of 18:09, 25 May 2020

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== 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=''&phi;φ'' = (1&nbsp;2&nbsp;9&nbsp;5&nbsp;4&nbsp;8&nbsp;6&nbsp;7&nbsp;3)}} and defining {{math|1=''x'' &sdot; ''y'' = ''&phi;φ''^''&minus;1''(''&phi;φ''(''x'') ∗ ''y'')}}.

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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''&nbsp;+&nbsp;2 |journal=Computing |volume=70 |issue=4 |pages=349–357 |issn=0010-485X |doi=10.1007/s00607-003-0017-3 }}</ref>

<ref name="damm2007">{{cite journal |last=Damm |year=2007 |first=H. Michael |title=Totally anti-symmetric quasigroups for all orders ''n''&ne;2≠2,6 |journal=Discrete Mathematics |volume=307 |issue=6 |pages=715–729 |issn=0012-365X |doi=10.1016/j.disc.2006.05.033 |doi-access=free }}</ref>

<ref name="fenwick2014">{{cite book |last=Fenwick |year=2014 |first=Peter |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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<ref group="lower-roman" name="BIS2003" >{{cite journal |year=2003 |author1=[http://www.math.md/people/beliavscaia-galina/ Beliavscaia Galina] |author2=[http://www.math.md/people/izbas-vladimir/ Izbaş Vladimir] |author3=[http://www.math.md/people/scerbacov-victor/ Şcerbacov Victor] |title=Check character systems over quasigroups and loops |journal=Quasigroups and Related Systems |volume=10 |issue=[http://www.math.md/en/publications/qrs/issues/v10–n1/ 1] |pages=[http://www.math.md/en/publications/qrs/issues/v10–n1/10592/ 1–28] |url=http://www.math.md/files/qrs/v10-n1/v10-n1-(pp1-28).pdf |issn=1561-2848 }} See page 23.</ref>

<ref group="lower-roman" name="Chen2009">{{cite book |author=Chen Jiannan |year=2009 |chapter=The NP-completeness of Completing Partial anti-symmetric Latin squares |pages=[http://www.academypublisher.com/proc/iwisa09/papers/iwisa09p322.htm 322–324] |chapter-url=http://www.academypublisher.com/proc/iwisa09/papers/iwisa09p322.pdf |title=Proceedings of 2009 International Workshop on Information Security and Application (IWISA 2009) |url=http://www.academypublisher.com/proc/iwisa09/ |publisher=Academy Publisher |isbn=978-952-5726-06-0 }} See page 324.</ref>

<ref group="lower-roman" name="Mileva2009">{{cite journal |date=2009 |last1=Mileva |first1=A. |last2=Dimitrova |first2=V. |title=Quasigroups constructed from complete mappings of a group (Z₂ⁿ,&oplus;⊕) |journal=Contributions, Sec. Math. Tech. Sci., MANU/MASA |volume=XXX |issue=1–2 |pages=75–93 |url=http://manu.edu.mk/contributions/NMBSci/Papers/2009_5_Mileva.pdf |issn=0351-3246 }} See page 78.</ref>

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