List decoding from erasures: bounds and code constructions

We consider the problem of list decoding from erasures. We establish lower and upper bounds on the rate of a (binary linear) code that can be list decoded with list size L when up to a fraction p of its symbols are adversarially erased. Such bounds already exist in the literature, albeit under the label of generalized Hamming weights, and we make their connection to list decoding from erasures explicit. Our bounds show that in the limit of large L, the rate of such a code approaches the "capacity" (1 - p) of the erasure channel. Such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate Ω(ε²/log(1/ε)) that can be efficiently list-decoded using lists of size O(1/ε) when an adversarially chosen (1 - ε) fraction of symbols are erased, for arbitrary ε > 0. This improves previous results in this vein, which achieved a rate of Ω(ε³log(1/ε)).