Universal Coding of Positive Integers with Suppressed Codeword Length Order

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This academic activity for four years was done on the universal coding of positive integers, as it is the coding which can effectively encode unbounded positive integers for any appearance probabilities. The results of this study can be summarized as follows :1.On the study of the universal coding which has good performance for relatively small positive integers(1)We proposed a code which groups the length of given positive integer with the geometric progression. It is asymptotically optimal. And it has the property of preserving length order, number order, and lexicographic order.(2)We proposed a code which groups the parameters of Affine Transformation for the fractal representation of images with the grouping strategy of positive integer coding. We show that the reversible fractal representation for images is possible.2.On the study of the universal coding which length is near known theoretically minimum length order(1)We proposed a code which uses numerical calculations for the assignment of the codewords. It enables that the order of the codeword length is the modified log-star function for the first time. For positive integer n, its time complexity for coding and decoding is both log n which is the theoretically minimum order.(2)We proposed a code whose each codeword component is different from Levenshtein-Elias code with one bit at most, where Levenshtein-Elias code is the most basic recursive-type positive integer code. The codeword length of the proposed code is shorter than log-star function in almost all of sufficiently large positive integers.