On communication with Perfect Feedback against Bit-flips and Erasures

We study the communication model with perfect feedback considered by Berlekamp (PhD Thesis, 1964), in which Alice wishes to communicate a binary message to Bob through a noisy adversarial channel, and has the ability to receive feedback from Bob via an additional noiseless channel. Berlekamp showed that in this model one can tolerate 1/3 fraction of errors (a.k.a., bit-flips or substitutions) with non-vanishing communication rate, which strictly improves upon the 1/4 error rate that is tolerable in the classical one-way communication setting without feedback. In the case when the channel is corrupted by erasures, it is easy to show that a fraction of erasures tending to 1 can be tolerated in the noiseless feedback setting, which also beats the 1/2 fraction that is maximally correctable in the no-feedback setting. In this thesis, we consider a more general perfect feedback channel that may introduce both errors and erasures. We show the following results:

1. If α, β ∈ [0, 1) are such that 3α + β < 1, then there exists a code that achieves a positive communication rate tolerating α fraction of errors and β fraction of erasures. Furthermore, no code can achieve a positive-rate in this channel when 3α + β ≥ 1.

2. For the case when 3α + β < 1, we compute the maximal asymptotic communication rate achievable in this setting.