# Quantum LDPC Codes Based on Cocyclic Block Matrices
## 1 Introduction  
The theoretical and experimental research on virtual quantum secure communication has gradually matured. However, due to the constraints of single photon sources, single photon detectors and the high cost of quantum devices, its practical application still faces many technical challenges. Extending the error correction coding technology in classical communication to the field of quantum information, the breakthrough in quantum error correction coding technology will undoubtedly become an important link in the practical application of quantum secure communication in the future, although quantum error-correcting codes (QECCs) are of only nominal significance at present in both quantum key distribution (QKD) and quantum secure direct communication (QSDC) [1; 2]. Therefore, QECCs play theoretically a vital role in quantum information fields owing to their effects in fighting decoherence in quantum computing large-scale data [3; 4; 5; 6; 7; 8; 9; 10]. The appeared errors from quantum channel noise and possible eavesdropper's interference can be corrected by good QECCs. These codes are useful for building quantum fault-tolerant circuits. In general, an important class of quantum codes called stabilizer codes is firstly proposed by [PERSON] [3]. Currently, in quantum coding theory, the stabilizer formalism of QECCs and its special form called Calderbank-Shor-Steane (CSS) are its main construction methods [4; 5; 6]. In quantum code construction theory, how to design the generator matrix resort to block matrix is the key. However, the error correction ability of short quantum codes is limited. The construction method of long-length quantum codes is becoming more important. In order to achieve high error correction ability, low-density parity check (LDPC) codes can achieve the maximum rate of reaching the Shannon limit with an error probability approaching zero when the codeword length tends to infinity. Since [PERSON] first proposed LDPC codes in the 1960s [11], two decades later, it caught the considerable interest of many researchers again [12; 13; 14; 15]. Subsequently, with the rapid development of quantum information, many research achievements in its quantum versions have been made in this field [16; 17; 18; 19; 20; 21; 22; 23]. However, compared to the in-depth study of classical LDPC codes, the corresponding quantum LDPC codes are still relatively less studied. One main obstacle is the difficulty of obtaining a more effective algorithm of iterative coding. With theapplication of large-scale data in quantum information fields, it becomes more important to encode massive data by making use of LDPC codes.  
In quantum coding theory, the main problem for constructing low-density quantum code is how to obtain the generator for the stabilizer. Based on the current research results in this field, we will further study the encoding methods of quantum long-codes stabilizers. In this paper, a class of quantum stabilizer codes spirited with a family of cocyclic matrices is proposed [24; 25]. The involved cocyclic block matrices are based on the Jacket matrix, of which the main property is that its inverse matrix can be gained by block-wise inverse [26; 27]. According to the properties, its physical decoder circuit can be realized relatively simply. Meanwhile, the constructed quantum codes also have no 4-cycles in their Tanner graphs, so their parity check matrices have sparse characteristics.  
The remainder of the paper is arranged as follows. In Section 2, we provide some necessary background and notations for QECCs. Then, in Sections 3 and 4, in terms of circulant permutation matrices, we investigate the construction of quantum codes generated from cocyclic block matrices which are applied to construct a generator matrix of quantum codes. Furthermore, its properties of low density are also analyzed in this section. Finally, some concluding remarks are drawn in Section 4.