  In this work we consider the communication of information in the presence of
an online adversarial jammer. In the setting under study, a sender wishes to
communicate a message to a receiver by transmitting a codeword x=x_1,...,x_n
symbol-by-symbol over a communication channel. The adversarial jammer can view
the transmitted symbols x_i 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. More generally, for a delay parameter 0<d<1, we study the scenario in
which the jammer's decision on the corruption of x_i must depend solely on x_j
for j < i - dn. In this work, we initiate the study of codes for online
adversaries, and present a tight characterization of the amount of information
one can transmit in both the 0-delay and, more generally, the d-delay online
setting. We prove tight results for both additive and overwrite jammers when
the transmitted symbols are assumed to be over a sufficiently large field F.
Finally, we extend our results to a jam-or-listen online model, where the
online adversary can either jam a symbol or eavesdrop on it. We again provide a
tight characterization of the achievable rate for several variants of this
model. The rate-regions we prove for each model are informational-theoretic in
nature and hold for computationally unbounded adversaries. The rate regions are
characterized by "simple" piecewise linear functions of p and d. The codes we
construct to attain the optimal rate for each scenario are computationally
efficient.
