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We consider geometrical optimization problems related to optimizing the errorprobability in the presence of a Gaussian noise. One famous questions in thefield is the "weak simplex conjecture". We discuss possible approaches to it,and state related conjectures about the Gaussian measure, in particular, theconjecture about minimizing of the Gaussian measure of a simplex. We alsoconsider antipodal codes, apply the \v{S}id\'ak inequality and establish sometheoretical and some numerical results about their optimality.

Optimality of codes with respect to error probability in Gaussian noise