On check character systems over quasigroups and loops

In this article we study check character systems that is error detecting codes, which arise by appending a check digit $a_n$ to every word $a_1a_2\dots a_{n-1}: a_1a_2\dots a_{n-1}\rightarrow a_1a_2\dots a_{n-1}a_n$ with the check formula $ (\dots((a_1\cdot\delta a_2)\cdot \delta^2a_3)\dots)\cdot \delta^{n-2}a_{n-1})\cdot\delta^{n-1}a_n=c$, where $Q(\cdot)$ is a quasigroup or a loop, $\delta$ is a permutation of $Q$, $c\in Q$. We consider detection sets for such errors as transpositions $(ab\rightarrow ba)$, jump transpositions $(acb\rightarrow bca)$, twin errors $(aa\rightarrow bb)$ and jump twin errors $(aca\rightarrow bcb)$ and an automorphism equivalence (a weak equivalence) for a check character systems over the same quasigroup (over the same loop). Such equivalent systems detect the same percentage (rate) of the considered error types.