Optimal Universal Coding of Integers

2026-10-01

AI Summary

This study addresses the long-standing open problem of determining the exact minimum expansion factor and optimal construction for universal codes over integers. By constructing a least favorable distribution family, this work proposes a core criterion for universal coding of integers (UCI) analogous to Kraft's inequality. Combining information-theoretic analysis, numerical computation, and rigorous mathematical proofs, it systematically establishes the theoretical bounds of the expansion factor. The primary contribution is the first precise determination of the minimum expansion factor as 2.000124757..., certified to fifteen decimal places, alongside the theoretical construction of an optimal code. These results provide both an exact characterization of the performance limits of universal codes and a novel analytical framework for future research in this domain.

Abstract

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.

Problem

• Universal coding of integers
• Optimal minimum expansion factor
• Worst-case ratio
• UCI inequality

Innovation

• Universal coding of integers
• Minimum expansion factor
• UCI inequality
• Least favorable distribution
• Optimal coding