Check Character Systems using Quasigroups: I

We consider check character systems with one check character over an arbitrary quasigroup under different check equations which have one permutation. The aim of using such a system is to discover transmission errors. These systems always detect all single errors (that is errors only one component of a code word). We establish necessary and sufficient conditions for a general quasigroup check character system over this quasigroup to be able to detect all transpositions. jump transpositions, twin errors and jump twin errors. We also study capabilities of check character systems over loops, in particular, over Moufang loops and groups when the permutation in the check equations is the identify permutation.Communicated by: P. Wild