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Damm algorithm

In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004, as a part of his PhD dissertation entitled Totally Antisymmetric Quasigroups.

14 ↔ 40, ..., c, x, y ∈ Q, the following implications hold:

  1. n is equivalent to the existence of a weak totally anti-symmetric quasigroup of order x ∗ x = 0 is needed. Such a quasigroup can be constructed from any totally anti-symmetric quasigroup by rearranging the columns in such a way that all zeros lay on the diagonal. And, on the other hand, from any weak totally anti-symmetric quasigroup a totally anti-symmetric quasigroup can be constructed by rearranging the columns in such a way that the first row is in natural order.

    Algorithm

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    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). It is useful if each main diagonal entry is 0.

  2. 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.
  3. The resulting interim digit gives the check digit and will be appended as trailing digit to the number.

Example

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The following operation table will be used. It may be obtained from the totally anti-symmetric quasigroup x ⋅ y = φ(φ(x) ∗ y).

Salomon, David (2005). Coding for Data and Computer Communications. Springer Science+Business Media, Inc. p. 36. ISBN 978-0387-21245-6.
  • 1 2 3 4 5 Damm, H. Michael (2003). "On the Existence of Totally Anti-Symmetric Quasigroups of Order 4k + 2". Computing. 70 (4): 349–357. doi:10.1007/s00607-003-0017-3. ISSN 0010-485X. S2CID 31659430.
    • {{cite journal |year=2003|last1=Beliavscaia|first1=Galina|last2=Izbaş|first2=Vladimir|last3=Şcerbacov|first3=Victor|title=Check character systems over quasigroups and loops |journal=Quasigroups and Related Systems |volume=10 |issue=1|pages=1–28|url=http://www.math.md/files/qrs/v10-n1/v10-n1-(pp1-28).pdf |issn=1561-2848}} See page 23.</ref>\n<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>\n<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 (Z<sub>2</sub><sup>n</sup>,⊕) |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>"}},"i":0}}]}'>