Computationally Efficient Codes for Adversarial Binary-Erasure Channels

We study communication models for channels with erasures in which the erasure pattern can be controlled by an adversary with partial knowledge of the transmitted codeword. In particular, we design block codes for channels with binary inputs with an adversary who can erase a fraction p of the transmitted bits. We consider causal adversaries, who must choose to erase an input bit using knowledge of that bit and previously transmitted bits, and myopic adversaries, who can choose an erasure pattern based on observing the transmitted codeword through a binary erasure channel with random erasures. For both settings we design efficient (polynomial time) encoding and decoding algorithms that use randomization at the encoder only. Our constructions achieve capacity for the causal and "sufficiently myopic"models. For the "insufficiently myopic"adversary, the capacity is unknown, but existing converses show the capacity is zero for a range of parameters. For all parameters outside of that range, our construction achieves positive rates.