This paper consists of three parts. The first part presents a large class of new
binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices
are constructed based on cyclic subgroups of finite fields. Experimental results show that
the codes constructed perform well over the binary-input AWGN channel with iterative
decoding using the sum-product algorithm (SPA). The second part analyzes the ranks of the
parity-check matrices of codes constructed based on finite fields with characteristic of 2
and gives combinatorial expressions for these ranks. The third part identifies a subclass
of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this
subclass with the SPA converges very fast.