On Extended Forney-Kovalev GMD decoding

Consider a code C with Hamming distance d. Assume we have a decoder ¿ that corrects ¿ errors and ¿ erasures if ¿¿ + ¿ ¿ d - 1, where a real number 1 < ¿ ¿ 2 is the tradeoff rate between errors and erasures for this decoder. This holds e.g. for l-punctured Reed-Solomon codes, i.e., codes over the field F q l of length n < q with locators taken from the subfield F q , where l ¿ {1, 2, . . .} and ¿ = 1+1/l. We propose an m-trial generalized minimum distance (GMD) decoder based on ¿. Our approach extends results of Forney and Kovalev (obtained for ¿ = 2) to the whole given range of ¿. We consider both fixed erasing and adaptive erasing GMD strategies. For l > 1 the following approximations hold. For the fixed erasing strategy the error correcting radius is ¿ F ¿ d/2 (1 - l -m /2). For the adaptive erasing strategy, ¿ A ¿ d/2 (1 - l -2m ) quickly approaches d/2 if l or m grows. The minimum number of decoding trials required to reach an error correcting radius d/2 is m A = 1/2 (log l d + 1). This means that 2 or 3 trials are sufficient to reach ¿ A = d/2 in many practical cases if l > 1.