A Continuous Projection Converse and a Weighted Construction for Uniquely Decodable Code Pairs

2026-09-27

AI Summary

This study investigates the problem of maximizing the sum-rate for uniquely decodable code pairs. Methodologically, it introduces a weighted completion gluing construction to achieve explicit codes with high sum-rates, and combines conditional entropy estimation with an iterative projection argument to establish theoretical upper bounds. The primary contributions include the construction of an explicit code of length 672 achieving a sum-rate exceeding 1.3186, alongside a rigorous proof that the corresponding upper bound is 1.4800. These results significantly advance the known bounds for this problem.

Abstract

We study the maximum sum rate of uniquely decodable code pairs. A weighted refinement of a complement-gluing construction yields an explicit code of length 672 and sum rate exceeding $1.318639029203$. An iterated projection argument, combined with a conditional entropy estimate and a justified continuous limit, gives an upper bound of $1.480063425539$.

Problem

• uniquely decodable code pairs
• maximum sum rate
• code construction
• upper bound

Innovation

• Uniquely Decodable Code Pairs
• Weighted Construction
• Continuous Projection
• Sum Rate
• Conditional Entropy