发表机构
National University of Defense Technology; Great Bay University（国防科技大学; 大湾区大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造最不利分布族并建立UCI不等式，精确确定了整数通用编码的最优最小扩展因子C*=2.000124757036101…，为构造最优UCI提供了理论基础。
AI 中文摘要
整数通用编码（UCI）为正整数提供二进制码字，使得对于每个非增信源分布 $P$，平均码字长度保持在 $K$ 倍 $\max\{1,H(P)\}$ 以内。最小的常数 $K$ 称为 UCI $\mathcal{C}$ 的最小扩展因子，记为 $C_{\mathcal{C}}^{*}$。最优最小扩展因子 $C^*=\inf\{C_{\mathcal{C}}^{*}\}$ 是对应于最优 UCI 的最小扩展因子。目前已知最优最小扩展因子位于区间 $2\le C^*\le 2.0386$。在本文中，我们构造了一族“一点加均匀尾部”分布，并证明对于每个通用码，最坏情况比率由该族中的某个分布达到，因此该族对 UCI 问题是最不利的。我们进一步建立了一个不等式，称为 UCI 不等式，它对 UCI 的作用与 Kraft 不等式对前缀码的作用相同：对于任何实数 $B$，它决定 $B$ 是低于还是高于 $C^*$。通过 UCI 不等式，我们获得了 $C^*$ 的等价定义。通过数值计算，我们确定 $C^*=2.000124757036101\cdots$，前十五位小数已得到认证。一旦 $C^*$ 已知，我们理论上可以构造最优 UCI。
英文摘要
Universal coding of integers (UCI) provides binary codewords for positive integers such that, for every nonincreasing source distribution $P$, the average codeword length stays within $K$ times $\max\{1,H(P)\}$. The smallest constant $K$ is called the minimum expansion factor of UCI $\mathcal{C}$, denoted $C_{\mathcal{C}}^{*}$. The optimal minimum expansion factor $C^*=\inf\{C_{\mathcal{C}}^{*}\}$ is the minimum expansion factor corresponding to the optimal UCI. The optimal minimum expansion factor is currently known to lie in the range $2\le C^*\le 2.0386$. In this paper, we construct a family of one-point plus uniform-tail distributions and prove that, for every universal code, the worst-case ratio is attained by a distribution in this family, so that 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 $B$, it decides whether $B$ lies below or above $C^*$. Through the UCI inequality, we obtain an equivalent definition of $C^*$. By numerical computation, we determine $C^*=2.000124757036101\cdots$, the first fifteen decimal digits being certified. Once $C^*$ is known, we can theoretically construct the optimal UCI.
