Codes Against Online Adversaries: Large Alphabets

In this paper, 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 <b>x</b>=(<i>x</i>₁,...,<i>x</i>_n) symbol-by-symbol over a communication channel. The adversarial jammer can view the transmitted symbols <i>x</i>_i one at a time and can change up to a <i>p</i>-fraction of them. However, for each symbol <i>x</i>_i, the jammer's decision on whether to corrupt it or not (and on how to change it) must depend only on <i>x</i>_j for <i>j</i> ≤ <i>i</i>. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge of <b>x</b>. More generally, for a delay parameter δ ∈ (0,1), we study the scenario in which the jammer's decision on the corruption of <i>x</i>_i must depend solely on <i>x</i>_j for <i>j</i> ≤ <i>i</i>-δ<i>n</i>. In this study, the transmitted symbols are assumed to be over a sufficiently large field F. The sender and receiver do not share resources such as common randomness (though the sender is allowed to use stochastic encoding). We present a tight characterization of the amount of information one can transmit in both the 0-delay and, more generally, the δ-delay online setting. We show that for 0-delay adversaries, the achievable rate asymptotically equals that of the classical adversarial model. For positive values of δ, we consider two types of jamming: additive and overwrite. We also extend our results to a jam-or-listen online model, where the online adversary can either jam a symbol or eavesdrop on it. We present computationally efficient achievability schemes even against computationally unrestricted jammers.