Universally optimal (wiretap) codes

2026-08-09

AI Summary

This study addresses the problem of achieving optimal secure communication over a wiretap channel with a noiseless main channel. For this model, the work establishes—for the first time—a connection between universally optimal codes and α-mutual information (with α=2). By minimizing a specific energy function and integrating tools from discrete geometric coding theory with information-theoretic analysis, the authors prove that such codes attain information-theoretic optimality for secure transmission over this channel. The results not only provide rigorous theoretical guarantees but also highlight the pivotal role of universally optimal codes in maximizing secrecy rates, thereby offering a novel perspective for the design of physical-layer security coding schemes.

Abstract

Universally optimal (UO) codes were introduced by H. Cohn and A. Kumar in 2007 and later extended to the discrete setting by Cohn and Y. Zhao. They minimize the ``energy'' among all codes of the same size for a certain class of potential functions. So far only a small number of specific functionals have been linked to information theory problems. We add one more, showing that UO codes optimize $α$-mutual information for $α=2$ in the context of wiretap channels with a noiseless main channel. This implies that UO codes are optimal for transmission over this type of wiretap channels.

Problem

• universally optimal codes
• wiretap channels
• α-mutual information
• information theory
• coding

Innovation

• universally optimal codes
• wiretap channels
• α-mutual information
• information theory
• energy minimization