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This paper is concerned with general analysis on the rank and row-redundancyof an array of circulants whose null space defines a QC-LDPC code. Based on theFourier transform and the properties of conjugacy classes and Hadamard productsof matrices, we derive tight upper bounds on rank and row-redundancy forgeneral array of circulants, which make it possible to consider row-redundancyin constructions of QC-LDPC codes to achieve better performance. We furtherinvestigate the rank of two types of construction of QC-LDPC codes:constructions based on Vandermonde Matrices and Latin Squares and givecombinatorial expression of the exact rank in some specific cases, whichdemonstrates the tightness of the bound we derive. Moreover, several types ofnew construction of QC-LDPC codes with large row-redundancy are presented andanalyzed.

Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes