We continue investigations of check character systems with one check character over quasigroups under check equations without a permutation. These systems always detect all single errors (i.e., errors in only one component of a code word) and can detect some other errors occuring during transmission of data. For construction of such systems we use totally conjugate orthogonal T-quasigroups. These quasigroups are isotopic to abelian groups and have six mutually orthogonal conjugate quasigroups. We prove that a check character system over any totally conjugate orthogonal T-quasigroup is able to detect all transpositions and twin errors and establish additional properties of a totally conjugate orthogonal T-quasigroup by which such system can detect all jump transpositions and all jump twin errors. Some models of totally conjugate orthogonal T-quasigroups which satisfy all of the required properties for detection of each of the considered types of errors and an information with respect to the spectrum of such quasigroups are given.