A Simple and Efficient Decoding Algorithm for Algebraic-Geometric Codes

• Main Authors: Chih-Wei Liu, 劉志尉
• Other Authors: Chung-Chin Lu
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/69141351180632628012

博士 === 國立清華大學 === 電機工程學系 === 87 === By adopting the extended syndrome matrix M with a restricted Gaussian elimination, a simple and efficient decoding algorithm for algebraic-geometric codes is developed. The decoding algorithm can be considered as the refinement of the Feng-Rao algorithm and can be implemented by a set of r parallel early stopped Berlekamp-Massey algorithms, where r is the smallest nonzero nongap of the algebraic-geometric curve over which the code is defined. The computation complexity of the algorithm is in the order of O(r n^2), which is the same as that of the Kotter''s algorithm, where n is the code length. Comparing with the Kotter''s algorithm, the proposed decoding algorithm is superior in the following aspects. Firstly, with the early stopped property, the proposed algorithm can save both processing time and computation complexity. In particular, for decoding (n, n-2t) BCH codes, i.e. r=1, the proposed algorithm requires only t+e iterations (or steps) to determine the error-locator polynomial, where e is the number of errors actually occurred. While, the Kotter''s algorithm requires the constant 2t iterations. Secondly, with storing both nonzero discrepancy as well as the corresponding coefficient vector, the proposed algorithm prevents from the additional multiplicative operations for the normalization of the saved coefficient vector. The saved coefficient vector needs to be normalized only when it is being used to update the currently used coefficient vector. And finally, an accurate method of counting the available candidates is developed in the algorithm. Based on the point of view from the Feng-Rao algorithm, the method to count the total number of the available candidates is not correct in that of the Kotter''s algorithm.