Abstract
Low-density parity-check (LDPC) codes have been shown to have near-capacity performance with iterative message-passing decoding and sufficiently long block length. Quasi-cyclic LDPC (QC-LDPC) codes are an important class of LDPC codes which can be encoded and decoded with low complexity and suitable for many applications. In practical systems, it is important to know the exact dimension of a QC-LDPC code since the dimension describes the number of protected information bits. Since the dimension is equal to the code length minus the rank of the parity-check matrix, finding the code dimension is equivalent to finding the rank of the parity-check matrix. Unfortunately, the parity check matrix for QC-LDPC codes is usually not full-rank, so we need some methods to compute the rank. In this thesis, we develop a new approach to calculate the exact rank of parity-check matrix for QC-LDPC codes based on the characteristic polynomials for circulant matrices. A formula for the rank of the parity-check matrices with only one row-block is first derived. We then extend the result to matrices with two or three row-blocks. These formulas are derived based on the technique that transforms the matrices into the upper-triangular form. Furthermore, a low-complexity algorithm is given to handle other types of matrices. Based on the obtained results, the rank of the parity-check matrix for QC-LDPC codes under good algebraic constructions can be easily obtained. We also demonstrate our results by some examples.