Generalized Universal Coding of Integers

Universal coding of integers (UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1, H( P)\}$ is within a constant factor, where H(P) is the Shannon entropy of the decreasing probability distribution P. However, if we consider the ratio of the expected codeword length to H(P), the ratio tends to infinity by using UCI, when H(P) tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized UCI, such that the ratio of the expected codeword length to H(P) is within a constant factor K. The definition of generalized UCI is proposed, and then the coding structure of generalized UCI is introduced. Finally, the asymptotically optimal generalized UCI is presented.