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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.<ref name="dhmd" />
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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).<ref name="dhmd" /> It is useful if each main diagonal entry is {{math|0}},<ref name=fenwick2014 /> because it simplifies the check digit calculation.
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