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Two of the most common models for channels with synchronisation errors arethe Binary Deletion Channel with parameter $p$ ($\text{BDC}_p$) -- a channelwhere every bit of the codeword is deleted i.i.d with probability $p$, and thePoisson Repeat Channel with parameter $\lambda$ ($\text{PRC}_\lambda$) -- achannel where every bit of the codeword is repeated $\text{Poisson}(\lambda)$times.  Previous constructions based on synchronisation strings yielded codes withrates far lower than the capacities of these channels [CS19, GL18], and theonly efficient construction to achieve capacity on the BDC at the time ofwriting 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 codesand use it to construct simple and efficient encoding and decoding algorithmsfor both channels with nearly optimal rates.

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