On Approximation, Bounding & Exact Calculation of Block Error Probability for Random Codes

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. Deviating from Shannon's 1959 solid angle argument, we project the problem into two dimensions and apply standard trigonometry. This enables us to also analyze Gaussian random codes in additive white Gaussian noise and binary random codes for the binary symmetric channel. We find that the Voronoi regions harden doubly-exponential in the blocklength and utilize that to propose the new median bound that outperforms Shannon's 1959 sphere packing hound for the uniform spherical ensemble, whenever the code contains more than three codewords. Furthermore, we propose a very tight approximation to simplify computation of both exact error probability and the two bounds.