Horstein, Burnashev, Shayevitz and Feder, Naghshvar et al. and others have studied sequential transmission of a k-bit message over the binary symmetric channel (BSC) with full, noiseless feedback using posterior matching. Yang et al. provide an improved lower bound on the achievable rate using martingale analysis that relies on the small-enough difference (SED) partitioning introduced by Naghshvar et al. SED requires a relatively complex encoder and decoder. To reduce complexity, this paper replaces SED with relaxed constraints that admit the small enough absolute difference (SEAD) partitioning rule. The main analytical results show that achievable-rate bounds higher than those found by Yang et al. (2021) are possible even under the new constraints, which are less restrictive than SED. The new analysis does not use martingale theory for the confirmation phase and applies a surrogate channel technique to tighten the results. An initial systematic transmission further increases the achievable rate bound. The simplified encoder associated with SEAD has a complexity below order$O(K^{2})$and allows simulations for message sizes of at least 1000 bits. For example, simulations achieve 99% of of the channel’s 0.50-bit capacity with an average block size of 200 bits for a target codeword error rate of$10^{-3}$.

Traditional communication systems transmit a code-word only after all message bits are available at the transmitter. This paper joins Guo & Kostina and Lalitha et al. in developing approaches for causal encoding, where the transmitter may begin transmitting codeword symbols as soon as the first message bit arrives. Building on the posterior matching encoders of Horstein, Shayevitz & Feder, and Naghshvar et al., this paper extends our computationally efficient systematic encoder to progressively encode using only the message bits that are causally available. Systematic codes work well with posterior matching on a channel with feedback, and they provide an immediate benefit when causal encoding is employed instead of traditional encoding. Our algorithm captures additional gains in the interesting region where the transmission rate$\mu$is higher than the source rate$\lambda$at which message bits become available. In this region, we improve performance further through the transmission of additional, non-systematic symbols before a traditional encoder would have even begun transmission.