Decoding algebraic-geometric codes beyond the error-correction bound
Mohammad Amin Shokrollahi, Hal Wasserman · 1998
Generalizing the high-noise decoding methods of [1, 19] to the class of algebraic-geometric codes, we design the first polynomialtime algorithms to decode algebraic-geometric codes significantly beyond the conventional error-correction bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constant-rate linear codes over a fixed field F q such that a codeword is efficiently, non-uniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the error-correction bound. 1 Introduction Error-correcting codes, originally designed to accommodate reliable transmission of information through unreliable ...