Improved Hamming sphere bounds on the MLD performance of binary linear codes

In this paper, improved Hamming sphere bounds on the maximum-likelihood decoding (MLD) performance of binary linear codes over additive white Gaussian noise (AWGN) channels are proposed. The proposed Hamming sphere bounds both on the frame and bit error probabilities are based on Gallager´s first bounding technique (GFBT), where the “good” region is chosen to be an Hamming sphere, which is different from the conventional definition by the Euclidean distance. The good region is then divided into small regions to tighten the union bound on the error probability caused by the Hamming sphere. The proposed bounds require only the knowledge of the truncated weight spectrum of the code, which is helpful when the whole weight spectrum is unknown or not computable. Numerical results show that the proposed bounds are tighter than most of the upper bounds and even tighter than the tangential-sphere bound (TSB) for high code rates.