  In this paper, we consider the problem of variable-length coding over the
class of memoryless binary asymmetric channels (BACs) with noiseless feedback,
including the binary symmetric channel (BSC) as a special case. In 2012,
Naghshvar et al. introduced an encoding scheme, which we refer to as the
small-enough-difference (SED) encoder, which asymptotically achieves both
capacity and Burnashev's optimal error exponent for symmetric binary-input
channels. Building on the work of Naghshvar et al., this paper extends the SED
encoding scheme to the class of BACs and develops a non-asymptotic upper bound
on the average blocklength that is shown to achieve both capacity and the
optimal error exponent. For the specific case of the BSC, we develop an
additional non-asymptotic bound using a two-phase analysis that leverages both
a submartingale synthesis and a Markov chain time of first passage analysis.
For the BSC with capacity $1/2$, both new achievability bounds exceed the
achievability bound of Polyanskiy et al. for a system limited to stop-feedback
codes.
