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

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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