Generalized representation of integers

Abstract

By the problem, called a representation of the integers, is meant the encoding of the positive integers without a known upper bound, for which unique decoding can be provided. From the viewpoint of information theory, this problem has been discussed theoretically in its relation to the universality, asymptotic optimality, and computation theory. The result in those approaches has been applied to the data compression scheme for the type of data referring to the past as well as the URR and other floating-pointnumber representations with variable exponential length. This paper discusses representation of the integers problem from the viewpoint of the codeword-length function, which is the function to represent the length of the codeword. As a result, a general construction method is presented which has the desirable property and can uniquely encode all representations. Several methods are given from this approach, which can construct not only the traditional representations of the integers but also the first representations of the integers which has not been considered. It is noted first that two construction methods of codeword length, which are independent of the encoding, are considered. One is based on the combination of the codeword-length functions, and the other is based on the codeword-length distribution. More precisely, it is shown that not only are the traditional representations derived, but also the representations that have not been considered in the past. Using detailed examples, it is shown that the computational complexity of the encoding and decoding depends on the codeword-length function.