On Approximation, Bounding Exact Calculation of Block Error Probability for Random Codes
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This paper presents a method to calculate the exact block error probability of random codes under maximum-likelihood decoding. The proposed method is applicable to a variety of channels. Particular focus is on both spherical and Gaussian random codes in additive white Gaussian noise as well as binary random codes on the binary symmetric channel. While for spherical random codes Shannon, in 1959, argued with solid angles in N-dimensional space, we project the problem into two dimensions and apply standard trigonometry. This simplifies the derivation and also allows for the analysis of Gaussian random codes which turn out to perform better for short blocklengths and high rates. For spherical random codes, we show how to efficiently evaluate the error probability with high precision. The difference to Shannon's sphere packing bound turns out to be small, as the Voronoi regions harden doubly-exponential in the blocklength. We show that, whenever the code contains more than three codewords, the sphere packing bound can be tightened by a new bound, that requires exactly the same effort to compute. Furthermore, we propose a very tight approximation to simplify computation of both exact error probability and the two bounds. For the binary symmetric channel, we derive an upper and a lower bound on the block error probability. The two only differ from the exact error probability, as for discrete alphabets there is a small, but nonzero probability to guess for the correct codeword, if several codewords have equal distance to the received word.
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