A Small Ring Of Quantum Ldpc Codes Constructed

Channel noise is a great obstacle that must to be overcome of information processing systems. Error-correcting codes can protect information against noise. The key idea is that if we wish to protect a message against the effects of noise, then we should encode the message by adding some redundant information to the message. That way, even if some of the information in the encoded message is corrupted by noise, there will be enough redundancy in the encoded message that it is possible to recover or decode the message so that all the information in the original message is recovered. Either classical error-correcting coding theory or quantum error-correcting coding theory is using this idea.Low-Density Parity-Check (LDPC) Codes have become the research focus for its outstanding performance in classical error-correcting coding fields. In all kinds of LDPC codes, quasi-cyclic LDPC (QC-LDPC) codes can be encoded in linear time using simple shift registers, and attract the attention of people. In quantum error-correcting coding fields, stabilizer codes is a richly structured class of codes with a close connection to classical error-correcting codes. Calderbank-Shor-Steane (CSS) codes is a kind of stabilizer codes with special structure. CSS codes can be constructed by a pair of binary linear codes with twisted condition, or a dual-containing binary linear code. A dual-containing LDPC code have cycles of length 4 (or 4-cycles, for short), which may cause fatal errors when a message-passing decoding algorithm is employed.In this dissertation, we firstly briefly introduce the history of the development of classical error-correcting coding theory and quantum error-correcting coding theory. We then introduce the basic knowledge of classic linear codes and quantum codes, which contains the basic definition and the basic conclusion of LDPC codes, QC-LDPC codes and CSS codes. We lastly show a construction of a pair of QC-LDPC codes with free 4-cycles and twisted relation. Therefor, CSS codes obtained by this pair of QC-LDPC codes have more advantages than based on a dual-containing LDPC code. In addition, we also show a combining method for exponent matrices of a pair of QC-LDPC codes, which inherit free 4-cycles and twisted relation from original codes, that is, the combined pair of QC-LDPC codes are free 4-cycles and satisfying twisted relation if the original pair of QC-LDPC codes are free 4-cycles and satisfying twisted relation. Applying the combining method, we can obtain CSS code with a wide range of parameters.