Abstract: Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.
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.