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

点击翻译按钮获取中文摘要

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