LDPC behaviour as density of parity-check matrix increases

Question

My assignment is to implement a Loopy Belief Propagation algorithm for Low-density Parity-check Code. This code uses a parity-check matrix H which is rather sparse (say 750-by-1000 binary matrix with an average of about 3 "ones" per each column). The code to generate the parity-check matrix is taken from here

Anyway, one of the subtasks is to check the reliability of LDPC code when the density of the matrix H increases. So, I fix the channel at 0.5 capacity, fix my code speed at 0.35 and begin to increase the density of the matrix. As the average number of "ones" in a column goes from 3 to 7 in steps of 1, disaster happens. With 3 or 4 the code copes perfectly well. With higher density it begins to fail: not only does it sometimes fail to converge, it oftentimes converges to the wrong codeword and produces mistakes.

So my question is: what type of behaviour is expected of an LDPC code as its sparse parity-check matrix becomes denser? Bonus question for skilled mind-readers: in my case (as the code performance degrades) is it more likely because the Loopy Belief Propagation algo has no guarantee on convergence or because I made a mistake implementing it?

Answer 1

After talking to my TA and other students I understand the following:

1. According to Shannon's theorem, the reliability of the code should increase with the density of the parity check matrix. That is simply because more checks are made.
2. However, since we use Loopy Belief Propagation, it struggles a lot when there are more and more edges in the graph forming more and more loops. Therefore, the actual performance degrades.
3. Whether or not I made a mistake in my code based solely on this behaviour cannot be established. However, since my code does work for sparse matrices, it is likely that the implementation is fine.