Upper bounds on the capacity of binary channels with causal adversaries

In this paper, 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=(x ₁,..., x_n) bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bits 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 a causal manner. Namely, for each bit x_i, the jammer's decision on whether to corrupt it or not must depend only on x_j for j ≤ i. This is in contrast to the 'classical' adversarial jamming situations in which the jammer has no knowledge of x, or knows x completely. In this study, we present upper bounds (that hold under both the average and maximal probability of error criteria) on the capacity which hold for both deterministic and stochastic encoding schemes.