On the tightness of dependence testing bound approximations in point-to-point memoryless channels

In this paper, upper and lower bounds on the dependence testing (DT) bound on the average decoding error probability in channel coding are derived. These bounds are obtained when the saddlepoint approximation is used to approach the DT bound, known for its computational complexity. The formulation is valid for any point-to-point memoryless channel configuration in a wide range of system parameters. These bounds allow to estimate the accuracy of the known normal and the new saddlepoint approximations, even when the fundamental limits are not known. The proposed bounds are simple to compute and provide an interesting addendum to the classical normal or saddlepoint approximations. We illustrate the interest of this approach for the binary symmetric channel and for the binary Gaussian channel. In all scenarios we studied, the bounds obtained from the saddlepoint approximation outperform the bounds obtained from the normal approximation. Index Terms-Finite block length, Berry-Esseen, saddlepoint approximation, binary symmetric channel.