Linear-time encodable/decodable codes with near-optimal rate

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a,fraction (1 - r - epsilon)/2 of errors over an alphabet of constant size depending only on E, for every 0 < r < 1 and arbitrarily small epsilon > 0. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are "near-MDS" codes). Such near-MDS linear-time codes were known for the decoding from erasures; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the Zyablov bound, and also the more general Blokh-Zyablov bound (by resorting to multilevel concatenation). Our work also yields linear-time encodable/decodable codes which match Forney's error exponent for concatenated codes for communication over the binary symmetric channel. The encoding/decoding complexity was quadratic in Forney's result, and Forney's bound has remained the best constructive error exponent for almost 40 years now. In summary, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in 1 addition have linear-time encoding and decoding algorithms.