Abstract
It is shown that i) erasures-and-errors decoding of Goppa codes can be done using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n \log^{2} n)</tex> arithmetic operations, ii) long primitive binary Bose-Chaudhuri-Hocquenghem (BCH) codes can be decoded using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n \log n)</tex> arithmetic operations, and iii) Justesen's asymptotically good codes can be decoded using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(n^{2})</tex> bit operations. These results are based on the application of efficient computational techniques to the decoding algorithms recently discovered by Sugiyama, Kasahara, Hirasawa, and Namekawa.
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