On a Relation of the Reliability Function and the Asymptotic Distance Ratio in Shannon's Channel Coding Theorem

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There are three main points of our research results as follows.1. We get an upper bound on the average probability of undetected error for the ensemble of binary expansions of generalized Reed-Solomon codes. From this bound, we simultaneously show that the asymptotic distance ration for the ensemble of these codes meets the Varshamov-Gilbert bound and this ensemble satisfies the expurgated bound.2. Iterated codes given by P. Elias are well known as a class of the codes satisfying the channel-coding theorem. By utilizing the way for deriving an upper bound on the average probability of undetected error for the ensemble of all binary systematic linear block codes, we can easily get that for the ensemble of all iterated codes. It is shown that the average capability for the ensemble of all iterated codes is poorer than that for all binary systematic linear block codes.3. Concatenated codes given by G. D. Forney, Jr. are very important codes from practical and theoretical viewpoint. By utilizing a characteristic structure of the concatenated code, an approximately good computation method of the probability of undetected error without knowing binary weight distribution of the concatenated code is proposed. Since the computational complexity of the method is at the most O(n) where n is a code length of a outer code of the concatenated code, it is an efficient method when investigating the capability of error detection for the concatenated code from practical and theoretical viewpoint. By comparing exact values with approximate values in some examples of the codes, which are small enough for the weight distribution to be found by computer search, we show the efficiency of the approximate values by the proposed method.