A recursive-type positive integer code is proposed. It prefixes the information about the length of the component of the codeword recursively. It is an asymptotically optimal code. The codeword length for a positive integer n is shorter than n bits in almost all of sufficiently large positive integers, where n is the log-star function.

A positive integer code EXEb,h,d(b1, h1,d0) is proposed. Its codeword for a positive integer n consists of three kinds of information: (1) how many times the number of n's digits can be subtracted by the terms of a progression including a geometric progression, (2) the rest of the subtractions, and (3) given value of the positive integer n. EXEb,h,d is a non-recursive type code. It is an asymptotically optimal code (for d1) and preserves the lexicographic,length, and number orders (for bh+2). Some examples of EXEb,h,d are also presented. Their codeword lengths are found to be shorter than the Amemiya and Yamamoto code CEk except for small positive integers.