Improved error probability bounds for block codes on the Gaussian channel

We derive eight upper and lower bounds on the soft decoding error probability for block codes of length n with M equally likely, equal-energy codewords c_i, i=0,..., M-1, represented in n-dimensional Euclidean space and received in the presence of additive white Gaussian noise. These bounds show significant improvement over well-known bounds, e.g., the union upper bound, Berlekamp´s tangential union bound, and Shannon´s sphere packing bounds. Following Shannon (1959), we consider the differentially thin conical shell d𝒮_n (Θ) between two circular cones of half-angles Θ and Θ+dΘ, each with vertex at the origin and axis passing through the correct codeword c₀