A characterization of the capacity of online (causal) binary channels

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x₁,... ,x_n) ∈ {0, 1}ⁿ bit by bit via a channel limited to at most pn corruptions. The channel is "online" in the sense that at the ith step of communication the channel decides whether to corrupt the ith bit or not based on its view so far, i.e., its decision depends only on the transmitted bits (x₁,... , x_i). This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has full knowledge of the transmitted codeword x. In this work we study the capacity of binary online channels for two corruption models: the bit-flip model in which the channel may flip at most pn of the bits of the transmitted codeword, and the erasure model in which the channel may erase at most pn bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of p. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization.