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This paper presents a simplified list-decoding algorithm to correct any number w of errors in any alternant code of any length n with any designed distance t+1 over any finite field Fq ; in particular, in the classical Goppa codes used in the McEliece and Niederreiter public-key cryptosystems. The algorithm is efficient for w close to, and in many cases slightly beyond, the Fq Johnson bound $J'=n'-\sqrt{n'(n'-t-1)}$ where n′=n(q−1)/q, assuming t+1≤n′. In the typical case that $qn/t\in (\lg n)^{O(1)}$ and that the parent field has $(\lg n)^{O(1)}$ bits, the algorithm uses $n(\lg n)^{O(1)}$ bit operations for $w\le J'-n/(\lg n)^{O(1)}$ ; O(n4.5) bit operations for $w\le J'+o((\lg n)/\lg\lg n)$ ; and nO(1) bit operations for $w\le J'+O((\lg n)/\lg\lg n)$ .