A Proof of the Weak Simplex Conjecture

06/23/2023
by   Adriano Pastore, et al.
0

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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset