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'''Theorem 1.''' ''Let y be a received word such that there exists a [http://en.wikipedia.org/wiki/Codeword codeword]'' <math>\mathbf{c} = (c_1,..., c_N) \in C_{out}\circ{C_{in}} \subseteq [q^n]^N</math> ''such that'' <math>\Delta(\mathbf{c}, \mathbf{y})</math> < <math>Dd \over 2</math>. ''Then the deterministic GMD algorithm outputs'' <math>\mathbf{c}</math>.
Note that a [http://en.wikipedia.org/wiki/Concatenated_codes naive decoding algorithm for concatenated codes] can correct up to <math>Dd \over 4</math> errors.
'''Lemma 1.''' ''Let the assumption in Theorem 1 hold. And if'' <math>\mathbf{y^{\prime\prime}}</math> ''has'' <math>e'</math> ''errors and'' <math>s'</math> ''erasures(when compared with'' <math>\mathbf{c}</math>'') after'' '''Step 1''', ''then'' <math>\mathbb{E}[2e' + s']</math> < <math>D</math>.
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