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: I am not quite sure what you mean with "rearranging the rows works" -- perhaps that the resulting quasigroup is still totally anti-symmetric? Well, I have not thought about your argument, but what I can say is that the table given on this Wikipedia page is not totally anti-symmetric, and hence the resulting checksum code does *not* detect all transposition errors. E.g. 41 and 14 both have checksum 1. While I am at it, I am confused as to why one would copy the quasigroup multiplication table from page 111 in Damm's thesis, which is inside a screenshot, instead of, say, taking the one on page 106... [[User:BlackFingolfin|BlackFingolfin]] ([[User talk:BlackFingolfin|talk]]) 08:49, 23 March 2015 (UTC)
:: After rearranging rows, the resulting quasigroup is ''weakly'' totally anti-symmetric. As you noticed 14 = 41, which violates the second requirement for TA-quasigroups. But notice that ''a''14 ≠ ''a''41 for all ''a'' ∈ ''Q'' and the algorithm described in the article uses a fixed prefix of ''a'' = 0 (interim digit initialized to 0), thus still detecting all adjacent transpositions. — [[User:MwGamera|mwgamera]] ([[User talk:MwGamera|talk]]) 16:18, 23 March 2015 (UTC)
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