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

本文的研究成果主要有以下三个方面。我们得到了广义Reed-Solomon码的二元展开式的系综的平均不可检错概率的上界。从这个界出发，我们同时证明了这些码的系综的渐近距离比满足Varshamov-Gilbert界，并且这个系综满足删除界。P.Elias给出的迭代码是众所周知的一类满足信道编码定理的码。通过利用推导出所有二进制系统线性分组码集合的平均不可检错概率的上界的方法，我们可以很容易地得到所有迭代码的集合的平均不可检错概率的上界。结果表明，所有迭代码集的平均性能比所有二进制系统线性分组码的平均性能都要差。级联码由G.D.Forney，Jr.从实践和理论的角度来看都是非常重要的规范。利用级联码的特征结构，在不知道级联码的二进制重量分布的情况下，提出了一种近似较好的不可检错概率的计算方法。由于该方法的计算复杂度最高为O(N)，其中n为级联码的外码长度，因此从实用和理论的角度研究级联码的检错能力是一种有效的方法。通过将精确值与某些码例的近似值进行比较，我们证明了该方法的近似值的有效性。

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.