I am new to working on LDPC codes and have gotten stuck on this issue for some time. I am struggling on how to deal with parity check matrices that are not of full rank. I generate the parity check matrix using Gallager's construction (see example 1.14 http://sigpromu.org/sarah/SJohnsonLDPCintro.pdf) and then attempt to get the result into systematic form, so that the matrix describes the information part and the parity part. When I do this however I am left with one or more all zero rows and to double check I checked the rank and it is indeed not full rank matrix.
The question I have is, what do I do when the parity check matrix starts out as not full rank? How do I get the systematic form and then the generator matrix? And what is the meaning of the parity check matrix not being full rank?