Capacity-Achieving Sequences for the Erasure Channel

Abstract — This paper starts a systematic study of capacityachieving 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 capacity-achieving in a much stronger sense than, e.g., the Tornado sequence. This also explains why numerical optimization techniques tend to favor graphs with only one degree of check nodes. Index Terms—Erasure channel, low-density parity-check codes, sparse graphs. I