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Design of BCH-Goppa Decoding Algorithms in Terms of the Tau-functions over Finite Fields

有限域上Tau函数的BCH-Goppa解码算法设计

基本信息

批准号：
09559011
负责人：
NAKAMURA Yoshimasa
金额：
$ 4.42万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

In 1997 the following results are given. To decrease the computational complexity of the algorithm based on moment problems introduced by the head of investigator of this project a new decoding algorithm is formulated by using the finite Toda molecule equation over finite fields, namely, the quotient difference algorithm over finite fields It is checked that the algorithm can be successfully applied to various examples including the Goppa code. Secondly, the dynamics of the finite Toda molecule equation over finite fields is considered. There exist conserved quantities and periodic orbits where each period is equal to the order of finite field or its divisor. The periods and the correspondence to the BCH-Goppa decoding are completely classified by the conserved quantities. Moreover, a relationship is discussed between the first component of the eigenvector of the Lax pair of the finite Toda molecule and the value of error appearing the BCH-Goppa code.In 1998, the results obtained in 19 … More 97 are published as a Phys. Lett. A paper entitled "Dynamics of the finite Toda molecule over finite fields and a decoding algorithm". It is shown that the modified KdV (mKdV) equation induces a continued fraction expansion which is different from that given by the Toda molecule. The mKdV equation describes a one-parameter deformation of the orthogonal polynomials named the symmetric Szego polynomials. An integrable discretization of the mKdV equation is found.In 1999, an integrable discretization of the Schur flow is given which is a generalization of the discrete mKdV equation and is corresponding to the general Szego polynomials. Hirota's bilinear form and the tau-function plays the central role. With the help of the discrete Schur flow a new O(N^2) algorithm for computing continued fraction expansions of given Herglotz class functions is designed Moreover, a new algorithm for solving algebraic equations is formulated by the discrete mKdV equation, which will be published in Inverse Problems. Less

1997年的结果如下：为了降低基于本项目负责人介绍的矩问题的译码算法的计算复杂度，利用有限域上有限Toda分子方程，提出了一种新的译码算法，即有限域上的商差算法，该算法可以成功地应用于包括Goppa码在内的各种实例。其次，考虑了有限Toda分子方程在有限场上的动力学性质。存在守恒量和周期轨道，其中每个周期等于有限域的阶数或它的除数。周期和与BCH-Goppa解码的对应关系由守恒量完全分类。此外，还讨论了有限Toda分子的Lax对的特征向量第一分量与出现BCH-Goppa码的误差值之间的关系。1998年，在19…中获得的结果，还有97个以物理学的形式发表。列托人。论文题目为“有限域上有限Toda分子的动力学和解码算法”。结果表明，修正后的KdV （mKdV）方程引起了与Toda分子不同的连分数展开。mKdV方程描述了一种称为对称Szego多项式的正交多项式的单参数变形。建立了mKdV方程的可积离散化。1999年，给出了舒尔流的可积离散化，它是离散mKdV方程的推广，与一般的Szego多项式相对应。Hirota的双线性形式和tau函数起着核心作用。利用离散Schur流，设计了一种计算给定Herglotz类函数的连分式展开式的O（N^2）新算法，并利用离散mKdV方程建立了求解代数方程的新算法，该算法将发表在《逆问题》杂志上。少