Low-density arrays of circulant matrices: Rank and row-redundancy, and QC-LDPC codes

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, tight bounds on rank and row-redundancy are derived, which make it possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better performance. Moreover, a new construction of QC-LDPC codes from random partitions of finite fields, which has flexible code dimensions and is abundant in row-redundancy, is presented and analyzed.