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A framework is presented for generalized minimum distance (GMD) decoding with a limited number of decoding trials and a restricted set of reliability values. In GMD decoding, symbols received from the channel may be erased before being fed into an algebraic error-erasure decoder for error correction, in subsequent or simultaneous trials with different erasing patterns. The decision whether or not to erase a symbol in a certain trial is taken by an erasure-choosing algorithm which takes into account reliability information from the channel. The final GMD decoder output is a codeword which results from a decoding trial and satisfies a certain distance criterion. For various erasing strategies and reliability sets, the guaranteed error-correction radius and the unsuccessful decoding probability of this technique are studied. Both known and new results, with applications to concatenated coding, follow from the unified approach presented in this correspondence.