Damm algorithm: Difference between revisions

Revision as of 03:53, 10 January 2013

Revision as of 03:54, 10 January 2013

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In [[error detection]], the '''Damm algorithm''' is a [[check digit]] [[algorithm]] that detects all [[Transcription error|single-digit [[checksumerrors]] errors and all [[Transcription error#Transposition Error|adjacent transposition]] errors]]. It was presented by H. Michael Damm in 2004. Its essential part is a [[quasigroup]] of [[Order (group theory)|order]] 10 (i.e. having a 10×10 [[Latin square]] as [[Cayley table|operation table]]) with the special feature of being totally anti-symmetric. Damm revealed several methods to create such TA-quasigroups of order 10 and gave some examples in his doctoral dissertation.<ref name=dhmd>Damm, H. Michael (2004). ''[http://archiv.ub.uni-marburg.de/diss/z2004/0516/pdf/dhmd.pdf Total anti-symmetrische Quasigruppen (Dr. rer. nat.).]'' Philipps-Universität Marburg.</ref> With this, Damm also disproved an old conjecture that TA-quasigroups of order 10 do not exist.<ref>Damm, H. Michael (2003). [http://link.springer.com/article/10.1007%2Fs00607-003-0017-3 "On the Existence of Totally Anti-Symmetric Quasigroups of Order 4''k''&nbsp;+&nbsp;2"] ''Computing'' '''70''' (4): 349–357.</ref>

== Algorithm ==

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== Strengths and weaknesses ==

The Damm algorithm is similar to the [[Verhoeff algorithm]]. It too will detect ''all'' occurrences of altering one single digit and ''all'' occurrences of transposing two adjacent digits. (These are the two most frequently appearing types of [[transcription error]]s.)<ref name=dhmd />). But the Damm algorithm has the benefit that it makes do without the dedicatedly constructed [[permutation]]s and its position specific [[Exponentiation#In_abstract_algebra|powers]] being inherent in the [[Verhoeff algorithm|Verhoeff scheme]]. Furthermore, a table of [[Inverse element|inverses]] can be dispensed with provided all diagonal entries of the operation table are zero.

The Damm algorithm does not suffer from exceeding the number of 10 possible values, resulting in the need for using a non-digit character (as the [[X]] in the [[ISBN#ISBN-10_check_digit_calculation|ISBN-10]] [[Check_digit#ISBN_10|check digit]] scheme).

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== References ==

<references />

*Damm, H. Michael (2007). [http://www.sciencedirect.com/science/article/pii/S0012365X06004225 "Totally anti-symmetric quasigroups for all orders ''n''&nbsp;!=&nbspne;2,6"] ''Discrete Mathematics'' '''307''' (6): 715–729.

{{DEFAULTSORT:Damm Algorithm}}