The strong simplex conjecture is false

The design of M average-energy-constrained signals in additive white Gaussian noise is addressed. The long-standing strong simplex conjecture, which postulates that the regular simplex signal set maximizes the probability of correct detection under an average-energy constraint, is disproven. A signal set is presented that performs better than the regular simplex signal set at low signal-to-noise ratios for all M⩾7. This leads to the result that, for all M⩾7, there is no signal set of M signals which is optimal at all signal-to-noise ratios. Furthermore, the optimal signal set at low signal-to-noise ratios is not an equal energy set for any M⩾7. The regular simplex is shown to be the unique signal set which maximizes the minimum distance between signals. It follows that a signal set which maximizes the minimum distance is not necessarily optimum. However, the regular simplex is shown to be globally optimum in the sense of uniquely maximizing the union bound on error probability at all signal-to-noise ratios