Binary Causal-Adversary Channels

In this work we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x=(x1,...,xn) bit-by-bit over a communication channel. The adversarial jammer can view the transmitted bits xi one at a time, and can change up to a p-fraction of them. However, the decisions of the jammer must be made in an online or causal manner. Namely, for each bit xi the jammer's decision on whether to corrupt it or not (and on how to change it) must depend only on x_j for j <= i. This is in contrast to the "classical" adversarial jammer which may base its decisions on its complete knowledge of x. We present a non-trivial upper bound on the amount of information that can be communicated. We show that the achievable rate can be asymptotically no greater than min{1-H(p),(1-4p)^+}. Here H(.) is the binary entropy function, and (1-4p)⁺ equals 1-4p for p < 0.25, and 0 otherwise.