Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs

TL;DR

This paper establishes tighter upper bounds on the size of one set in unbalanced uniquely decodable code pairs, improving previous bounds especially when the sets are nearly balanced.

Contribution

It introduces sharper bounds for unbalanced UDCPs, significantly improving upon earlier results for small epsilon values.

Findings

Upper bound on beta: 0.4228 + sqrt(epsilon)

Improved bounds for small epsilon compared to prior work

Applicable to unbalanced code pairs with large sets

Abstract

Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| = 2^{(1-\epsilon)n}$ and $|B| = 2^{\beta n}$, satisfies $\beta \leq 0.4228 +\sqrt{\epsilon}$. For sufficiently small $\epsilon$, this bound significantly improves previous bounds by Urbanke and Li~[Information Theory Workshop '98] and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound $\beta$ by $0.4921$ and $0.4798$, respectively, as $\epsilon$ approaches $0$.