  Quantum key distribution (QKD) is a popular protocol that provides
information theoretically secure keys to multiple parties. Two important
post-processing steps of QKD are 1) the information reconciliation (IR) step,
where parties reconcile mismatches in generated keys through classical
communication, and 2) the privacy amplification (PA) step, where parties
distill their common key into a new secure key that the adversary has little to
no information about. In general, these two steps have been abstracted as two
distinct problems. In this work, we consider a new technique of performing the
IR and PA steps jointly through sampling that relaxes the requirement on the IR
step, allowing for more success in key creation. We provide a novel LDPC code
construction known as Block-MDS QC-LDPC codes that can utilize the relaxed
requirement by creating LDPC codes with pre-defined sub-matrices of full-rank.
We demonstrate through simulations that our technique of sampling can provide
notable gains in successfully creating secret keys.
