This paper is concerned with general analysis on the rank and row-redundancy of an
array of circulants whose null space defines a QC-LDPC code. Based on the Fourier transform
and the properties of conjugacy classes and Hadamard products of matrices, we derive tight
upper bounds on rank and row-redundancy for general array of circulants, which make it
possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better
performance. We further investigate the rank of two types of construction of QC-LDPC codes:
constructions based on Vandermonde Matrices and Latin Squares and give combinatorial
expression of the exact rank in some specific cases, which demonstrates the tightness of
the bound we derive. Moreover, several types of new construction of QC-LDPC codes with
large row-redundancy are presented and analyzed.