  For an arbitrary degree distribution pair (DDP), we construct a sequence of
low-density parity-check (LDPC) code ensembles with girth growing
logarithmically in block-length using Ramanujan graphs. When the DDP has
minimum left degree at least three, we show using density evolution analysis
that the expected bit-error probability of these ensembles, when passed through
a binary erasure channel with erasure probability $\epsilon$, decays as
$\mathcal{O}(\exp(-c_1 n^{c_2}))$ with the block-length $n$ for positive
constants $c_1$ and $c_2$, as long as $\epsilon$ is lesser than the erasure
threshold $\epsilon_\mathrm{th}$ of the DDP. This guarantees that the coset
coding scheme using the dual sequence provides strong secrecy over the binary
erasure wiretap channel for erasure probabilities greater than $1 -
\epsilon_\mathrm{th}$.
