Approach to Some Open Problems about Shannon's Channel Coding Theorem

我们的研究结果有以下四点。设A （w, w_<inf>）为Hamming权值w= w_<inf>+w_<che>的信息符号权值w_<inf>和检查符号权值w_<che>的码字数，则用递归式得到显式A （w, w_<inf>）。利用最大距离可分离码的Hamming权分布的某些特征结构和系统码字的命题，我们可以得到权枚举数来计算广义Reed-Solomon码的二进制展开的未检测错误概率的上界和下界。为了显示这些权重枚举员的有效性，本报告中计算了数值。即给出了计算里德-所罗门码经过计算机所有搜索后二进制展开的权值分布的具体参数，并计算出该码出现未检测到错误概率的真实值。此外，对于给定的n个Reed-Solomon码的具体参数，利用一类广义Reed-Solomon码的平均权值分布，计算了广义Reed-Solomon码的二进制展开式集合上出现未检测到错误的平均概率。通过将这些值与由这些权重枚举数计算的广义Reed-Solomon码的二进制展开的未检测错误概率的上界和下界值进行比较，表明了这些权重枚举数的有效性。3 .利用最大距离可分离码的Hamming权分布的某些特征结构和系统码字的命题，我们得到了计算一类具有广义Reed-Solomon外码的连接码二进制展开的未检出错误概率的上下界的权枚举子。通过使用Justesen码的特征结构，Kolev和Kohnosu—Tokiwa明确给出了低码率Justesen码族的权重分布。然而，这些论文并没有给出这些族的渐近评价。然后，这些族的渐近距离比的收敛点不是基于下界，而是基于从这些权值分布中得到的最小权值。通过与Justesen码的渐近距离比下界的比较，明确了这些族具有渐近能力的位置。少

There are four in points of our research results as follows1.Let A (w, w_<inf>) be the number of codewords of both weight w_<inf> in the information symbols and weight w_<che> in the check symbols for the Hamming weight w= w_<inf>+w_<che>, then we got the explicit A (w, w_<inf>) with a recurrence formula.2.By utilizing certain characteristic structure of the Hamming weight distribution of maximum distance separable codes and proposition of systematic codewords, we can get weight enumerators to calculate upper and lower bounds on the probability of an undetected error for binary expansions of generalized Reed-Solomon codes. To show the effectiveness of these weight enumerators, the numerical value is calculated in this report. That is, concrete parameters that can calculate the weight distribution of binary expansion of Reed-Solomon code by all searches of the computer are given, and the true value of the probability of an undetected error for this code is calculated. Moreover, for give … More n concrete parameters of Reed-Solomon codes, the average probability of an undetected error over the set of binary expansions of generalized Reed-Solomon codes is calculated by using the average weight distribution for a class of generalized Reed-Solomon codes. By comparing these values with values of upper and lower bound on the probability of an undetected error for binary expansions of generalized Reed-Solomon codes calculated from those weight enumerators, the effectiveness of those weight enumerators is shown.3.By utilizing certain characteristic structure of the Hamming weight distribution of maximum distance separable codes and proposition of systematic codewords, we can get weight enumerators to calculate upper and lower bounds on the probability of an undetected error for a class of binary expansions of concatenated codes with generalized Reed-Solomon outer codes.4.By using a feature structure of the Justesen code, the weight distributions for the families of the Justesen codes having low code rates are explicitly given by Kolev and Kohnosu--Tokiwa. However, an asymptotic evaluation to these families is not given in those papers. Then, the convergent points of the asymptotic distance ratio that those families have are specified on the basis of not a lower bound but minimum weights obtained from those weight distributions. Comparing with the lower bound on the asymptotic distance ratio for Justesen codes, the location of the asymptotic ability that those families have is clarified. Less