Cycle Analysis and Construction of Protographs for QC LDPC Codes With Girth Larger Than 12

A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2
i
,
i
= 6,7,8,9,10, i.e., the cycles existing regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2
g,

g
ges 2, its protograph code cannot have the inevitable cycles of length smaller than 6
g
. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14.