Abstract
A simple tight upper bound, without any integral and optimal operations in its final version, is proposed over additive white Gaussian noise (AWGN) channels. We derive the simple bound both on the frame probability and on the bit error probability. The proposed bound is within the framework of Gallager’s first bounding technique (GFBT), in which the Gallager region is chosen by the hamming distance, avoiding the limitation of the Euclidean distance. To compute the upper bound inside the Gallager region, we can tighten the union bound not only by collecting more information about the receiving vector which can cause the maximum-likelihood (ML) decoding error events, but also by employing the independence between the ML decoding error events and some positions of the received vectors. The proposed bound is tight since numerical results show that it is tighter than the proposed bound by Divsalar (without any integral and optimal operations) and the proposed bound by Poltyrev (considered as one of the tightest upper bounds for the binary linear codes with lower rate).
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