Total anti-symmetrische Quasigruppen

Totally anti-symmetric quasigroups are employed in check digit systems. Until today their existence for all orders $4k+2\geq 10$ was unsettled. Ecker and Poch conjectured in 1986 that there are no totally anti-symmetric quasigroups of order $4k+2$. We disprove this conjecture and develop constructions for totally anti-symmetric quasigroups of order $n$ for all $n\neq 2,6$. By a computer search we prove in addition that check digit systems over a 2-quasigroup of the order 10, just as check digit systems over groups of order 10, cannot detect all (jump) twin errors or jump transpositions. As a further result we show that the class of totally anti-symmetric quasigroups is no variety.