A Proof of the Weak Simplex Conjecture

06/23/2023

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by   Adriano Pastore, et al.

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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.