Conjectured tighter lower bound on the mapping probability ρ(i)

Establish whether, under the generalized requirement that for every integer k with 1 ≤ k ≤ |S| there exist at least k coded symbols each containing at most k source symbols in the Rateless IBLT bipartite graph, the mapping probability ρ(i)—defined as the probability that a random source symbol maps to the i-th coded symbol—must satisfy the asymptotic lower bound ρ(i) = ω(1/i).

Background

In the design of Rateless IBLT, the mapping probability ρ(i) specifies how likely a random source symbol is mapped to the i-th coded symbol. Decodability of the peeling decoder imposes constraints on how fast ρ(i) can decrease with i. Lemma 1 shows that to ensure the presence of at least one pure coded symbol within the first m coded symbols with non-negligible probability, ρ(i) cannot decrease slower than nearly 1/i, i.e., ρ(i) = Ω(1/i^{1−ε}) for any ε > 0.

The authors suggest a generalization of Lemma 1 that requires at least k coded symbols to have at most k source symbols mapped to them, for every k ≤ |S|. They conjecture that this stronger requirement would imply an even tighter bound on ρ(i), specifically ρ(i) = ω(1/i). Proving or refuting this conjecture would clarify the tightness of the necessary conditions on ρ(i) for successful peeling in Rateless IBLT.