A Simple Method to Calculate Random-Coding Union Bounds for Ultra-Reliable Low-Latency Communications

Ultra-Reliable Low-Latency Communications are based on very short codes whose performances cannot be evaluated correctly by using the Shannon capacity formula holding for asymptotically large code lengths. The use of Random-Coding Union Bounds (RCUB's) has been suggested in the literature as an alternative for this application. Unfortunately, their calculation is difficult and the Gaussian approximation may lead to erroneous results. The saddlepoint approximation has been proposed as an alternative to overcome the limitations of the Gaussian approximation. Though this technique is valid in many cases, situations exist where the exact calculation provides different results. A simple numerical technique is proposed in this letter to calculate numerically the exact value of the RCUB. Its accuracy is compared to that of the Gaussian and saddlepoint approximations in some cases of interest.