A Proof of the Weak Simplex Conjecture

Adriano Pastore

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arXiv · 2306.13478

A Proof of the Weak Simplex Conjecture

Adriano Pastore

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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.

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Adriano Pastore. 2023-06-23. A Proof of the Weak Simplex Conjecture. https://arxiv.org/abs/2306.13478

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