The Construction of Near-optimal Universal Coding of Integers

The Universal Coding of Integers~(UCI) is suitable for discrete memoryless sources with unknown probability distributions and infinitely countable alphabet sizes. A UCI is a class of prefix codes for which the ratio of the average codeword length to $\max\{1,H(P)\}$ is within a constant expansion factor \textcolor{red}{$C_{\mathcal{C}}$} for any decreasing probability distribution $P$, where $H(P)$ is the entropy of $P$. For any UCI code $\mathcal{C}$, \emph{the minimum expansion factor} \textcolor{red}{$C_{\mathcal{C}}^{*}$} is defined to represent the infimum of the set of extension factors of $\mathcal{C}$. Each $\mathcal{C}$ has a unique corresponding \textcolor{red}{$C_{\mathcal{C}}^{*}$}, and the smaller \textcolor{red}{$C_{\mathcal{C}}^{*}$} is, the better the compression performance of $\mathcal{C}$ is. The class of UCIs $\mathcal{C}$ (or a family $\{\mathcal{C}_i\}_{i=1}^{\infty}$) that achieves the smallest \textcolor{red}{$C_{\mathcal{C}}^{*}$} is defined as the \emph{optimal UCI}. The best current result is that the range of $C_{\mathcal{C}}^{*}$ for the optimal UCI is $2\leq C_{\mathcal{C}}^{*}\leq 2.5$. In this paper, we prove a tighter probability inequality for decreasing distributions, which serves as a new tool for studying the properties of UCIs. On the basis of this inequality, we prove that there exists a class of near-optimal UCIs, called the $\nu$ code, achieving \textcolor{red}{$C_\nu=2.0386$}. This narrows the range of the minimum expansion factor for the optimal UCI to $2\leq C_{\mathcal{C}}^{*}\leq 2.0386$. We show that the $\nu$ code is currently optimal in terms of the minimum expansion factor. In addition, we propose a new proof showing that the minimum expansion factor of the optimal UCI is lower bounded by $2$.