Optimality of codes with respect to error probability in Gaussian noise

TL;DR

This paper explores geometric optimization problems to minimize error probability in Gaussian noise, discusses the weak simplex conjecture, and investigates antipodal codes' optimality using theoretical and numerical methods.

Contribution

It presents new approaches and conjectures related to the weak simplex conjecture and Gaussian measure minimization, and analyzes antipodal codes' optimality.

Findings

Discussion of approaches to the weak simplex conjecture

Conjectures about minimizing Gaussian measure of a simplex

Numerical results supporting antipodal codes' optimality

Abstract

We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the "weak simplex conjecture". We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the \v{S}id\'ak inequality and establish some theoretical and some numerical results about their optimality.