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. Our results 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 that can be eciently list decoded using lists of size O(1=") from up to a fraction (1 ") of erasures, for arbitrary " > 0. This improves previous results from [19] in this vein, which achieved a rate of log(1="))

Ideal error-correcting codes: Unifying algebraic and number-theoretic algorithms

Over the past five years a number of algorithms decoding some well-studied error-correcting codes far beyond their “error-correcting radii” have been developed. These algorithms, usually termed as listdecoding algorithms, originated with a list-decoder for Reed-Solomon codes [36, 17], and were soon extended to decoders for Algebraic Geometry codes [33, 17] and as also some number-theoretic codes [12, 6, 16]. In addition to their enhanced decoding capability, these algorithms enjoy the benefit of being conceptually simple, fairly general [16], and are capable of exploiting soft-decision information in algebraic decoding [24]. This article surveys these algorithms and highlights some of these features