In this paper, we develop a new approach for rank analysis of parity-check matrices for quasi-cyclic low-density parity-check codes based on the associated polynomials of circulant matrices, applicable to general finite fields and arbitrary circulant sizes. Some formulas on the rank for parity-check matrices with one, two, and three row-blocks are first derived. For the general case with arbitrary numbers of row-blocks, lower and upper bounds on the rank are presented, and these bounds can be combined to give the exact rank result under certain conditions. We also investigate the effect on the rank by changing the circulant size. Furthermore, we study the relations between the rank of the masked matrix and that of the masking matrix, which can be used to predict the rank of the parity-check matrix after masking. The obtained rank analysis results are then applied to several classes of existing algebraically constructed parity-check matrices, along with their masked matrices. Finally, we demonstrate how rank analysis can be used in the code design procedure.

Quasi-cyclic low-density parity-check (QC-LDPC) codes are an important class of LDPC codes which can be encoded and decoded with low complexity and suitable for many applications. As the code dimension, which describes the number of protected information bits, is equal to the code length minus the rank of the parity-check matrix and the parity-check matrix for QC-LDPC codes is usually not full-rank, determining the rank of the parity-check matrix is of essential importance. In this paper, we study the rank of the parity-check matrix for QC-LDPC codes based on the associated polynomials for circulant matrices. A formula for the rank of the parity-check matrix with only one row-block is first derived. We then extend the result to matrices with two, three, or more row-blocks. Some bounds are also presented for matrices with arbitrary numbers of row-blocks. Furthermore, the exact rank is determined for a class of algebraically constructed parity-check matrices.