Explicit and Efficient Construction of (nearly) Optimal Rate Codes for Binary Deletion Channel and the Poisson Repeat Channel

Two of the most common models for channels with synchronisation errors are the Binary Deletion Channel with parameter $p$ ($\text{BDC}_p$) -- a channel where every bit of the codeword is deleted i.i.d with probability $p$, and the Poisson Repeat Channel with parameter $\lambda$ ($\text{PRC}_\lambda$) -- a channel where every bit of the codeword is repeated $\text{Poisson}(\lambda)$ times. Previous constructions based on synchronisation strings yielded codes with rates far lower than the capacities of these channels [CS19, GL18], and the only efficient construction to achieve capacity on the BDC at the time of writing this paper is based on the far more advanced methods of polar codes [TPFV21]. In this work, we present a new method for concatenating synchronisation codes and use it to construct simple and efficient encoding and decoding algorithms for both channels with nearly optimal rates.