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

我们的研究成果有以下四点：1.设A(W)为信息符号中的权数w和校验符号中的权数w的个数，则得到了显式的A(w，w；2.利用最大距离可分码汉明重量分布的某些特征结构和系统码字的命题，得到重量枚举器，用于计算广义Reed-Solomon码二进制展开的不可检错误概率的上下界。为了显示这些权重枚举器的有效性，本报告计算了该数值。也就是说，给出了计算Reed-Solomon码在所有搜索下的二进制展开的重量分布的具体参数，并计算了该码的不可检测错误概率的真值。此外，为了给…对于更多具体的Reed-Solomon码参数，利用一类广义Reed-Solomon码的平均重量分布，计算了在广义Reed-Solomon码的二进制展开集上不可检测错误的平均概率。3.通过利用最大距离可分码汉明重量分布的某些特征结构和系统码字的命题，我们可以得到加权枚举器来计算一类具有广义里德-所罗门外码的级联码的二进制展开的不可检测错误概率的上下界。4.利用Justesen码的特征结构，Kolv和Kohnosu-Tokiwa明确给出了具有低码率的Justesen码族的重量分布。然而，这些文献并没有给出对这些族的渐近估计。然后，这些族的渐近距离比的收敛点不是基于由这些权分布得到的下界而是最小权来指定的。与Justesen码的渐近距离比的下界相比较，阐明了这些族的渐近能力的位置。较少

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