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

本文的研究结果有以下四个方面：1.设A（w，w_<inf>）为<inf><che>汉明重量w= w_ +w_时，信息码和校验码<inf>的重量均为w_的码字个数，利用递推公式<che>得到了A（w，w_<inf>）的显式表达式; 2.利用最大距离可分码的汉明重量分布的某些特征结构和系统码字的命题，得到了计算广义Reed-Solomon码二进制展开式不可检错误概率上、下界的重量计数器.为了显示这些权重枚举器的有效性，本报告中计算了数值。也就是说，通过计算机的所有搜索，给出可以计算Reed-Solomon码的二进制扩展的权重分布的具体参数，并且计算该码的未检测到错误的概率的真实值。此外，对于Give 关于我们 利用一类广义Reed-Solomon码的平均重量分布，在Reed-Solomon码的具体参数n的基础上，计算了广义Reed-Solomon码的二元展开集上的平均不可检错误概率.通过与由这些重量计数器计算的广义Reed-Solomon码的二元展开的不可检错误概率的上、下界值的比较，说明了这些重量计数器的有效性。3.利用最大距离可分码的Hamming重量分布的某些特征结构和系统码字的命题，我们可以得到权重枚举器来计算具有广义Reed-Solomon外码的级联码的一类二进制扩展的不可检测错误概率的上界和下界。4.通过使用Justesen码的特征结构，Kolev和Kohnosu-Tokiwa明确给出了具有低码率的Justesen码族的权重分布。然而，这些文件中没有给出这些家庭的渐近评价。然后，这些家庭的渐近距离比的收敛点的基础上指定的不是一个下界，但从这些重量分布获得的最小重量。通过与Justesen码的渐近距离比的下界比较，明确了这类码族的渐近能力的位置。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