Capacity-Achieving Sequences for the Erasure Channel

This paper starts a systematic study of capacity-achieving sequences of low-density parity-check codes for the erasure channel. We introduce a class A of analytic functions and develop a procedure to obtain degree distributions for the codes. We show various properties of this class which will help us to construct new distributions from old ones. 

We then study certain types of capacity-achieving sequences and introduce new measures for their optimality. For instance, it turns out that the right-regular sequence is capcity-achieving in a much stronger sense then, e.g., the Tornado sequence. This also explains why numerical optimization techniques tend to favor graphs with only one degree of check nodes. 

Using our methods, we attack the problem of reducing the fraction of degree 2 variable nodes, which has important practical implications. It turns out that one can produce capacity achieving sequences for which this fraction reamins below any constant, albeit at the price of slower convergence to capacity.