Optimal Universal Coding of Integers
Wei Yan, Yunghsiang S. Han, Leqian Zheng
原文摘要
Universal coding of integers (UCI) provides prefix codes for positive integers such that, for every nonincreasing source distribution $P$, the average codeword length under $P$ stays within $K$ times $\max\{1,H(P)\}$. For a given UCI $\mathcal{C}$, the smallest such constant $K$ is called the minimum expansion factor of $\mathcal{C}$, denoted $C_{\mathcal{C}}^{*}$. Let $C^{*}\triangleq \inf \{C_{\mathcal{C}}^{*}\}$ over all UCIs. A UCI attaining $C^{*}$, or a family of UCIs attaining $C^{*}$ in the limit, is called optimal. It is currently known that $2\le C^{*}\le 2.0386$. In this paper, we construct a family of one-point plus uniform-tail distributions and prove that, for every UCI, a distribution in this family attains the worst-case ratio; thus, the family is least favorable for the UCI problem. We further establish an inequality, called the \emph{UCI inequality}, which plays the same role for UCI as the Kraft inequality does for prefix codes: for any real number $B\ge 1$, it decides whether $B$ lies below or above $C^*$. The UCI inequality gives an equivalent definition of $C^*$. By numerical computation, we determine $C^* = 2.00012475703610197861701176297579886254\cdots$, the first $38$ decimal digits being certified. Once $C^*$ is known, we can theoretically construct the optimal UCI.
cs.IT