Abstract: A systematic and exhaustive method based on the group
structure of a unitary Lie algebra is proposed to generate an enormous
number of quantum codes. With respect to the algebraic structure,
the orthogonality condition, which is the central rule of generating
quantum codes, is proved to be fully equivalent to the distinguishability
of the elements in this structure. In addition, four types of
quantum codes are classified according to the relation of the codeword
operators and some initial quantum state. By linking the unitary Lie
algebra with the additive group, the classical correspondences of some
of these quantum codes can be rendered.