Project/Area Number  02650262 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
情報工学

Research Institution  Toyohashi University of Technology 
Principal Investigator 
SAKATA Shojiro Toyohashi University of Technology, Department of KnowledgeBased Information Engineering, Professor, 工学部, 教授 (20064157)

Project Fiscal Year 
1990 – 1991

Project Status 
Completed(Fiscal Year 1991)

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1991 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1990 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  ErrorCorrecting Codes / Algebraic Geometry Codes / Algebraic Curves / Computational Complexity of Decoding Algorithm / Fast Decoding Algorithm / TwoDimensional BerlekampMassey Algorithm / 2D Syndrome Array / Total Ordering over the 2D Integral Lattice / 誤り訂正符号 / 代数幾何学符号 / 代数曲線 / 復号の計算量 / 高速復号法 / 2次元BerlekampーMasseyアルゴリズム / 2次元シンドロム配列 / 2次元整数格子上の全順序 / 2次元BerlekampーMasseyアリゴリズム 
Research Abstract 
Recently a sequence of errorcorrecting codes that have good errorcorrection 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 BerlekampMassey algorithm. The 2D BerlekampMassey algorithm which we proposed before solves th e 2D version of the problem treated by the (1D) BerlekampMassey 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.
