Performance of optimal codes at nite length

We consider the simple additive white Gaussian noise (AWGN) channel. In his 1959 paper [1], Claude E. Shannon showed that an optimal code is built by uniformly placing codewords on an n-dimensional sphere. An upper and a lower bound for the word error rate performance Pew of such a spherical code have been established by Shannon on an AWGN channel [1] for finite n. The main code parameters are its length n and its information rate R. The length n is the number of real dimensions. The information rate R is expressed in bits per real dimension. The spherical code is an ensemble of 2 points uniformly placed on a sphere in IR. A quick review of Shannon results and its generalization to a Rayleigh fading channel can be found in [7]. For n ≥ 100, the upper and lower bounds of Pew are superimposed. Hence, an accurate approximation for Pew is its lower bound Q(θ0), the probability of a codeword being moved outside its cone of half-angle θ0. Before you read [1] and [7], let me summarize all the numerical evaluations by two formulas. The first one is used to find θ0 from n and R, the second one to evaluate Q(θ0), where G = 1 2 [√