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.

翻译：我们解决了关于$n$维欧氏空间中满足码字能量约束、由$n+1个码字构成的码本在最小化平均译码错误概率意义下的最优码本结构这一长期未决的公开问题。该猜想断言：最优码本由内接于单位球面的正则单纯形（正四面体在$n$维空间中的推广）的$n+1$个顶点构成。本文基于对称性论证，并采用松弛方法（即联合优化码本与判决区域，而非仅优化码字位置）对该猜想给出了自包含的证明。