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 an to every word a1a2...an−1 : a1a2...an−1 → a1a2...an−1an with the check formula (...((a1 · δa2) · δ 2a3)...) · δ n−2an−1) · δ n−1an = c, where Q(·) is a quasigroup or a loop, δ is a permutation of Q, c ∈ Q. We consider detection sets for such errors as transpositions (ab → ba), jump transpositions (acb → bca), twin errors (aa → bb) and jump twin errors (aca → 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.