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维欧几里得空间中由n+1个码字组成的码本在码字能量约束下，以最小化平均译码错误概率为目标的码本最优结构的长期未解问题。该猜想指出：最优码本由内接于单位球面的正单纯形（正四面体在n维空间的推广）的n+1个顶点构成。本文给出了该猜想的自包含证明，其核心思想基于对称性论证，并采用松弛方法——即联合优化码本与决策区域，而非仅优化码字位置。