The Equality Cases of the Weak Simplex Conjecture

2026-08-19

AI Summary

研究解决了在加性白高斯噪声下，n+1个等概率等能量信号在R^n中如何排列以最大化正确解码概率的问题，证明了正则单纯形是唯一最优解。

Abstract

Among $n+1$ equiprobable equal-energy signals in $\R^n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.

Problem

• signal arrangement
• correct decoding probability
• regular simplex
• signal-to-noise ratio
• Shannon

Innovation

• regular simplex
• maximum-likelihood decoding
• additive white Gaussian noise
• correct-decoding probability
• machine-checked