Some Comments on the Strong Simplex Conjecture

TL;DR

This paper revisits the disproof of the Strong Simplex Conjecture, clarifies the proper interpretation of signal-to-noise ratio, and confirms the conjecture remains false with a revised counterexample that aligns with classical SNR definitions.

Contribution

It corrects the interpretation of SNR used in prior disproofs and provides a modified counterexample consistent with the Channel Coding Theorem.

Findings

The original counterexample's SNR interpretation was flawed.

A new signal set outperforms the regular simplex at low SNR.

The Strong Simplex Conjecture remains disproven under proper SNR interpretation.

Abstract

In the disproof of the Strong Simplex Conjecture presented in [Steiner, 1994], a counterexample signal set was found that has higher average probability of correct optimal decoding than the corresponding regular simplex signal set, when compared at small values of the signal-to-noise ratio. The latter was defined as the quotient of average signal energy and average noise power. In this paper, it is shown that this interpretation of the signal-to-noise ratio is inappropriate for a comparison of signal sets, since it leads to a contradiction with the Channel Coding Theorem. A modified counterexample signal set is proposed and examined using the classical interpretation of the signal-to-noise ratio, i.e., as the quotient of average signal energy and average noise energy. This signal set outperforms the regular simplex signal set for small signal-to-noise ratios without contradicting the Channel Coding Theorem, hence the Strong Simplex Conjecture remains proven false.