The Simplex Conjecture
Posted: Fri Dec 19, 2014 10:31 pmThis problem was suggested by Prof. Max Costa.
The (weak) simplex conjecture in communication theory states that the best signal set with n+1 vectors in R^n for transmission over an AWGN channel under the constraint that all vectors have the same norm is the regular simplex centered at the origin.
The strong simplex conjecture stated that this was also true under an average energy constraint. However, the strong simplex has been disproved. It has been shown by Steiner that there is a signal with n+1 = 7 vectors that performs better than the regular simplex. Nevertheless, the weak version of the conjecture remains open.
Please see the .pdf files attached for more information.
The (weak) simplex conjecture in communication theory states that the best signal set with n+1 vectors in R^n for transmission over an AWGN channel under the constraint that all vectors have the same norm is the regular simplex centered at the origin.
The strong simplex conjecture stated that this was also true under an average energy constraint. However, the strong simplex has been disproved. It has been shown by Steiner that there is a signal with n+1 = 7 vectors that performs better than the regular simplex. Nevertheless, the weak version of the conjecture remains open.
Please see the .pdf files attached for more information.