Efficient Decoding Method of Some Algebraic Geometry Codes

点击翻译按钮获取中文摘要

Recently a sequence of error-correcting codes that have good error-correction performance and high coding rate have been discovered using algebraic geometry and ideas of a russian coding theorist V. D. Goppa. These codes are a class of algebraic codes which are derived from algebraic curves over finite fields and are called algebraic geometry codes. Several investigators in Europe as well as in Japna have proposed decoding methods for some algebraic geometry codes quite recently. The computational complexties in most of these decoding methods are of the order of n^3 or more, where n is the code length. On the other hand, some investigators gave a fast algorithm by applying our 2D Berlekamp-Massey algorithm. The 2D Berlekamp-Massey algorithm which we proposed before solves th e 2D version of the problem treated by the (1D) Berlekamp-Massey algorithm, that is synthesis of 2D linear feedback shift register capable of generating a given finite 2D array. Both these algorithms in principle have the same computational complexity of the order of n^2, where n is the size of the given 1D or 2D array in this case. In the progression of this research we have revealed some more applications of our algorithm to decode several other algebraic geometry codes by making clear some novel aspects of the algorithm and by giving some new versions of it. We have shown that our methods can be applied to decode efficiently several new algebraic geometry codes which just have been discovered by other coding theorists. The computational complexities of our decoding methods are less than those of the previous methods. Furthermore, we have given several extensions of our algorithm to apply them to decode some other codes which have not been constructed yet but have the possibility of being invented.