On check character systems over quasigroups and loops

Abstract

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 \(a_1a_2...a_{n-1}: a_1a_2...a_{n-1} \rightarrow a_1a_2...a_{n-1}a_n\) with the check formula \( (...((a_1\cdot \delta a_2)\cdot \delta^2a_3)...)\cdot \delta^{n-2}a_{n-1})\cdot\delta^{n-1}a_n = c\), 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.