Design of near-optimum quantum error-correcting codes based on generator and parity-check matrices of LDGM codes

We study the design of near-optimum quantum error correcting codes based on the use of sparse matrices. The basic idea is to construct a Calderbank-Shor-Steane (CSS) code based on the generator and parity-check matrices of a classical channel code with low density generator matrix (LDGM code), which is designed with a specific structure inspired in the parallel concatenation of regular LDGM codes. Then, row operations are performed in both matrices to achieve the desired quantum rate. Decoding is performed in an iterative manner, by applying message passing over the corresponding graphs. The proposed codes allow greater flexibility and are easier to design than existing sparse-graph quantum codes, while leading to better performance.