Totally anti-symmetric quasigroups for all orders n≠2,6n≠2,6

A quasigroup (Q,*)(Q,*) is called totally anti-symmetric if (c*x)*y=(c*y)*x⇒x=y(c*x)*y=(c*y)*x⇒x=y and x*y=y*x⇒x=yx*y=y*x⇒x=y. A totally anti-symmetric (TA) quasigroup can be used for the definition of a check digit system. Ecker and Poch [Check character systems, Computing 37 (1986) 277–301] conjectured that there are no TA-quasigroups of order 4k+24k+2. This article will completely disprove their conjecture (except for n=2,6n=2,6) as we will give constructions for TA-quasigroups for all orders n≠2,6n≠2,6. Additionally we prove that the class of TA-quasigroups is no variety.