Block-MDS QC-LDPC Codes with Application to High-Dimensional Quantum Key Distribution

High-dimensional quantum key distribution (QKD) is a popular protocol that provides information theoretically secure keys to multiple parties. Two important 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 design our IR protocol to be aware of the PA step and utilize sampling to relax the requirement on the IR step without sacrificing the final key length of the PA step, allowing for more bits generated in key creation utilizing practical decoders. We provide a novel PA-aware LDPC code construction known as Block-MDS QC-LDPC codes that can utilize the relaxed requirement. We demonstrate through simulations that our technique of sampling can provide notable gains in successfully creating secret keys.