Fast Decodicng Method of Any One-Point Algebraic-Geometric Codes up to the Feng-Rao Bound

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We have established a fast decoding algorithm of any one-point algebraic-geometric (AG) code, which is defined on an arbitrary algebraic curve, up to the Feng-Rao designed distance. For the codelength n, our method has computational complexity of order O (n^<7/>) and of order less than O (n^3) to decode a one-point AG code defined on a Hermitian plane curve and on any algebraic curve of higher dimension, respectively, while the fundamental Feng-Rao method has complexity of order O (n^3). In regard to efficiency, out method is the best among all the known decoding methods of any one-point AG code. These results were presented in part at the 1994 IEEE Int. Symp. Inform. Theory, at the 1994 IEEE Int. Worshop Inform. Theory, and at some other conferences. The contents were published partially in Finite Fields and Their Applications, Vol.1,1995, and the main part will appear in the forthcoming Special Issue of IEEE Trans. Inform. Theory. The details of our theory were published in Bullet. U … More nv.Elect. -Comm., Vol.8,1995. In parallel to the above theoretical work, we made some computer experiment. We implemented a software system (C-program) for our decoding method, and by applying it to two kinds of codes defined on a Hermitian plane curve and on its three-dimensional extension, we investigated the actual efficiency of our method. As a result of simulation on many random error-patterns, it was shown that both the number of arithmetics over the finite field and the actual computing time have a tendency quite similar to the theoretical computational complexity, which gives an additional evidence to our theory and a guideline for practical use in future. Furthermore, it was made sure that our method can decode beyond the designed distance in some cases. On the other hand, we published a paper containing a general review on a broader class of AG codes in Bullet. Jap.Soc.Ind.Appl.Math., Vol.4,1994. In addition, we have proceeded to investigate parallel processor architecture for hardware implementation of our method and fast error-and-erasure decoding as extensions of the present research. Some results of the research were presented at the AAECC-11 Conference and at the 1995 IEEE Int.Symp.Inform.Theory, and in some other conferences. Less