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

1997年给出了以下结果。为了降低基于矩问题的译码算法的计算复杂度，利用有限域上的有限TODA分子方程，提出了一种新的译码算法，即有限域上的商差分译码算法。该算法可以成功地应用于包括Goppa码在内的各种例子。其次，研究了有限域上有限个Toda分子方程的动力学问题。存在守恒量和周期轨道，其中每个周期等于有限域或其因子的阶数。BCH-Goppa译码的周期和对应关系完全由守恒量来分类。此外，还讨论了有限Toda分子Lax对的特征向量的第一分量与BCHGoppa码出现的误差值之间的关系.1998年，19…的结果更多的97本书以物理杂志的形式出版。让我们来吧。一篇题为《有限域上有限Toda分子的动力学和译码算法》的论文。结果表明，修正的KdV(MKdV)方程引起了与Toda分子不同的连分式展开。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。给出了mKdV方程的一个可积离散形式，并在1999年给出了Schur流的一个可积离散形式，它是离散mKdV方程的推广，对应于一般的Szego多项式.Hirota的双线性形式和tau函数起着核心作用。利用离散Schur流，设计了一个计算给定Hergltz类函数的连分式展开式的新的O(N^2)算法，并利用即将发表在反问题中的离散mKdV方程给出了一个求解代数方程的新算法.较少

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