Classical self-orthogonal codes and their applications to quantum codes

• Main Author: Jin, Lingfei
• Other Authors: Xing Chaoping
• Format: Theses and Dissertations
• Language: English
• Published: 2013
• Subjects: DRNTU::Science::Mathematics::Algebra; DRNTU::Science::Mathematics::Applied mathematics
• Online Access: http://hdl.handle.net/10356/52463
• Institution: Nanyang Technological University

After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. One of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. Classical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. Another good candidate of classical codes to investigate for dual containment comes from the class of algebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. After various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.