Shannon's channel coding problems from points of view of both information theory and algebraic coding theory

已知二元线性分组码的可靠性函数、渐近距离比和不可检错误概率的各界都已给出。因此，我们认为，对于一些重要的二元线性分组码子类的集合，求出这些函数的各个上界，对于解决信息论或纠错码理论中的一些悬而未决的问题是一个重要的研究课题。(1)通过使用Justesen码的特征结构，这些家庭的渐近距离比的收敛点被指定的基础上，而不是一个下限，但从这些重量分布获得的最小重量。(2)利用最大距离可分码的Hamming重量分布的某些特征结构，我们可以得到重量计数器，用于计算广义Reed-Solomon码的二进制扩展的不可检测错误概率的上界和下界.此外，未检测到的错误的平均概率的值通过使用所有里德-所罗门码的所有码字的二进制扩展的集合的平均权重分布对于一些给定的具体代码参数来计算。(3)本文给出了GF（2 m）上Wozencraft的随机移位码系综中所有码的重量计数器的一些性质

It is well known that each bound of reliability function, asymptotic distance ratio, and the probability of undetected error for the ensemble of all binary linear block codes is given. Therefore we think that it is an important research to get each bound of those functions for an ensemble of some important subclasses of binary linear block codes in order to find a clue to a solution for some open problems in information theory or the theory of error correcting codes. (1) By using a feature structure of the Justesen code, 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. (2) By utilizing certain characteristic structure of the Hamming weight distribution of maximum distance separable codes, we can get weight enumerators to compute an upper and a lower bound on the probability of an undetected error for binary expansions of generalized Reed-Solomon codes. Also, values of the average probability of an undetected error are computed by using the average weight distribution for an ensemble of binary expansions of all codewords of all Reed-Solomon codes for some given concrete code parameters. (3) We show some properties of the weight enumerators of all the codes over GF (2m) in Wozencraft's ensembles of randomly shifted codes