A Proof of the Weak Simplex Conjecture

TL;DR

This paper proves the long-standing weak Simplex Conjecture, establishing that the optimal codebook for certain Euclidean space codes with minimal error probability is a regular simplex, using symmetry and relaxation techniques.

Contribution

It provides a self-contained proof of the weak Simplex Conjecture, a key open problem in coding theory, by introducing a novel relaxation approach and symmetry arguments.

Findings

Optimal codebooks are regular simplexes.

Proof leverages symmetry and relaxation methods.

Confirms conjecture for codes with n+1 codewords in n-dimensional space.

Abstract

We solve a long-standing open problem about the optimal codebook structure of codes in $n$-dimensional Euclidean space that consist of $n+1$ codewords subject to a codeword energy constraint, in terms of minimizing the average decoding error probability. The conjecture states that optimal codebooks are formed by the $n+1$ vertices of a regular simplex (the $n$-dimensional generalization of a regular tetrahedron) inscribed in the unit sphere. A self-contained proof of this conjecture is provided that hinges on symmetry arguments and leverages a relaxation approach that consists in jointly optimizing the codebook and the decision regions, rather than the codeword locations alone.