Jump codes are a special class of quantum error correcting codes. The authors study combinatorial and symmetry properties of these codes. The paper considers the quantum correction model where errors are detected due to quantum jumps. This means that, for example a state of \(| 1>\) is spontaneously transformed into a state of \(| 0>\). A quantum operator is defined and then it is used to analyze symmetries with respect to usage of group representations. Section 4 presents the main result which is theorem 4, this provides a theoretic setting for construction of jump codes. A variant of this construction is then presented in section 5 and section 6 deals with further examples.