  A simple and general definition of quasi cyclic low density parity check (QC
LDPC) codes which are constructed based on circulant permutation matrices (CPM)
is proposed. As an special case of this definition, we first represent one type
of so called combinatorially designed multiple edge protograph codes. The code
construction is mainly based on perfect difference families (PDF) and is called
Construction 1. Secondly, using the proposed Construction 1 along with a
technique named as column dispersion technique (CDT), we design several types
of multiple-edge CPM QC LDPC codes (codes with Construction 2) in a wide range
of rates, lengths, girths and minimum distances. Parameters of some of these
codes are reported in tables. Also included in this paper are the
multiplicities of short simple cycles of length up to 10 in Tanner graph of our
constructed codes. Our experimental results for short to moderate length codes
show that although minimum distances of codes play an important role in
waterfall region, the higher the number of short simple cycles is, the better
(sharper) the waterfall is. The performances of many codes such as WiMAX, PEG,
array, MacKay, algebraic and combinatorial, and also, symmetrical codes have
compared with our constructed codes. Based on our numerical results and
regardless of how a code is constructed, those with higher number of short
simple cycles and higher minimum distances, have better waterfalls.
