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

Abstract: 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.