Book ; Online: A Proof of the Weak Simplex Conjecture
2023
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. ...
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. Comment: 7 pages, submitted for peer review |
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Keywords | Computer Science - Information Theory ; Mathematics - Metric Geometry |
Publishing date | 2023-06-23 |
Publishing country | us |
Document type | Book ; Online |
Database | BASE - Bielefeld Academic Search Engine (life sciences selection) |
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