New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes

Abstract

In this paper, new techniques are presented to either simplify or improve most existing upper bounds on the maximum-likelihood (ML) decoding performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectrum by Ma et al is re-derived in a detailed way based on Gallager's first bounding technique (GFBT), where the "good region" is specified by a sub-optimal list decoding algorithm. The error probability caused by the bad region can be upper-bounded by the tail-probability of a binomial distribution, while the error probability caused by the good region can be upper-bounded by most existing techniques. Secondly, we propose two techniques to tighten the union bound on the error probability caused by the good region. The first technique is based on pair-wise error probabilities. The second technique is based on triplet-wise error probabilities, which can be upper-bounded by the fact that any three bipolar vectors form a non-obtuse triangle. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the Q-function. The proposed bounds can also be adapted to bit-error probabilities. Index TermsThis work was presented in part at ISIT'2011.

Upper Bound on the Bit Error Probability of Systematic Binary Linear Codes via Their Weight Spectra

“…e Gallager Region R. We define the region R by the Hamming distance based on a list decoding algorithm which is shown in Figure 1, resulting in an irregular high-dimensional geometry (Algorithm 1). e list decoding algorithm is similar to but different from the algorithm presented in [14]. e list region in [14] is an n-dimensional Hamming sphere with center at the hard decision of the whole received sequence, while the list region here is a k-dimensional Hamming sphere with center at the hard decision of the information part of the received sequence.…”

“…Upper bounds on the maximum a posteriori (MAP) decoding error probability, as a key technique for evaluating the performance of the binary linear codes over additive white Gaussian noise (AWGN) channels, bring a profound impact on the reliable transmission of the next-generation mobile communication systems since they can be used to not only predict the performance without resorting to computer simulations but also guide the design of coding [1]. In order to improve the looseness of the union bound in the low signal-to-noise ratio (SNR) region, many improved upper bounds, on the bit error probability [2][3][4][5] and on the frame error probability [2][3][4][6][7][8][9][10][11][12][13][14][15], are proposed. As surveyed in [1], the improved upper bounds on the bit error probability [2][3][4] are based on Gallager's first bounding technique (GFBT):…”

Block Markov Superposition Transmission: Construction of Big Convolutional Codes From Short Codes

“…Then A(X, Y ) can be calculated recursively by performing a forward trellis-based algorithm [61] over the polynomial ring in Algorithm 4. Given A(X, Y ), the upper bound for the BER of the BMST system can be calculated by an improved union bound [62].…”

“…Fortunately, as shown in [62], truncated IOWEF suffices to give a valid upper bound, a fact that can be used to simplify the computation by removing certain states from the trellis.…”

On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

“…A complete generators' pseudo-weight distribution for the BCH[31,21,5] code of HBCH[31,21] with 627,052,479 generators.…”