On extended fomey-kovalev GMD decoding

Consider a code C with Hamming distance d. Assume we have a decoder Φ that corrects ε errors and θ erasures if λε+θ ≤ d - 1, where a real number 1 < λ ≤ 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 ε {1, 2, ⋯} and λ = 1+1/l. We propose an m-trial generalized minimum distance (GMD) decoder based on Φ. Our approach extends results of Forney and Kovalev (obtained for λ = 2) to the whole given range of λ. 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 PF ≈ d/2(1 - l^-m/2). For the adaptive erasing strategy, PA ≈ 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 PA = d/2 in many practical cases if l > J.