In this paper, amended truncated union bounds on the bit error probability of systematic linear codes over additive white Gaussian noise (AWGN) channels are proposed. The proposed bounds are based on Gallager's first bounding technique (GFBT), where the “good” region is defined by a suboptimal list decoding algorithm. The proposed bounds require only the knowledge of partial input output weight enumerating function (IOWEF) with truncated information weight, which can also be adapted to the codes when the IOWEF is not available and difficult to calculate but the weight spectrum is known. The proposed amended truncated union bounds improve the conventional union bounds and the recently proposed Ma bounds but have a similar complexity since they involve only the Q-function.
In this paper, amended truncated union bounds on the bit error probability of systematic linear codes over additive white Gaussian noise (AWGN) channels are proposed. The proposed bounds are based on Gallager's first bounding technique (GFBT), where the “good” region is defined by a suboptimal list decoding algorithm. The proposed bounds require only the knowledge of partial input output weight enumerating function (IOWEF) with truncated information weight, which can also be adapted to the codes when the IOWEF is not available and difficult to calculate but the weight spectrum is known. The proposed amended truncated union bounds improve the conventional union bounds and the recently proposed Ma bounds but have a similar complexity since they involve only the Q-function.
In this paper, upper bound on the probability of maximum a posteriori (MAP) decoding error for systematic binary linear codes over additive white Gaussian noise (AWGN) channels is proposed. The proposed bound on the bit error probability is derived with the framework of Gallager’s first bounding technique (GFBT), where the Gallager region is defined to be an irregular high-dimensional geometry by using a list decoding algorithm. The proposed bound on the bit error probability requires only the knowledge of weight spectra, which is helpful when the input-output weight enumerating function (IOWEF) is not available. Numerical results show that the proposed bound on the bit error probability matches well with the maximum-likelihood (ML) decoding simulation approach especially in the high signal-to-noise ratio (SNR) region, which is better than the recently proposed Ma bound.
The theoretical performance of binary codes for the additive white Gaussian noise (AWGN) channel is discussed. In particular, Gallager’s coding theorem for binary codes is treated extensively. By assuming a binomial weight distribution for linear codes, it is shown that the decoder error probability performance of some of the best, known linear, binary codes is the same as the average performance of the ensemble of all randomly chosen, binary nonlinear codes having the same length and dimension. Assuming a binomial weight distribution, an upper bound is determined for the erasure performance of any code, and it is shown that this can be translated into an upper bound for code performance in the AWGN channel. Finally, theoretical bounds on the construction of error-correction codes are discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
New lower bounds on the error probability of block codes with maximum-likelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen's lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binary-symmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin (1998) and by Keren and Litsyn (2001). Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caen-based bounds.
In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of binary linear codes transmitted over an additive white Gaussian noise channel. The bounding technique is based on the Hunter bound, which is a second-order upper bound in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight distribution, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are densely distributed in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density affects the improvement of the proposed bound over the conventional LP union bound. This paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code.
A new tight upper bound on the maximum-likelihood (ML) word and bit-error decoding probabilities for specific codes over discrete channels is presented. It constitutes an enhanced version of the Gallager upper bound and its variations resulting from the Duman-Salehi second bounding technique. An efficient technique is developed that, in the case of symmetric channels, overcomes the difficulties associated with the direct computation of the proposed bound. Surprisingly, apart from the distance and input-output weight enumerating functions (IOWEFs), the bound depends also on the coset weight distribution of the code.
With the development of communication technology, modern communication networks need to achieve ultra-high reliability to meet practical demands. Forward error correction coding has been widely used in real-world communication systems for providing reliable data transmission over noisy channels. In this thesis, we investigate the efficient performance evaluation methods for block codes over memoryless channels. Performance bounds are powerful tools to provide insights on the error performance of the coded systems. In many practical scenarios where performance bounds are not applicable (e.g., due to the unavailability of the relevant coding parameters under a given decoder), the Monte Carlo simulation is still popular despite its inefficiency, especially in the low error probability regime. As a variance-reduction technique, importance sampling (IS) can significantly reduce the sample size needed for reliable estimation. We firstly examine the problem of efficient performance evaluation of linear block codes over binary symmetric channels (BSCs). Contrary to the conventional wisdom, we prove that for BSCs, all bounds based on Gallager’s first bounding technique, including the famous union bound, are not asymptotically tight for all possible choices of the Gallager region. By proposing the so-called input demodulated-output weight enumerating function (IDWEF) of a code, asymptotically tight maximum-likelihood decoding upper and lower bounds for BSCs are derived. For the cases that coding parameters are absent or suboptimal decoders are applied, we propose an efficient IS estimator by deriving the optimal IS distribution of the Hamming weight of the error vector. In addition, the asymptotic percentage saving of the proposed IS estimator over the state-of-the-art IS estimator is characterized in terms of the sample size. Its accuracy in predicting the efficiency of the proposed IS estimator is verified by extensive computer simulation. We further examine the IS estimator design and efficiency analysis problems for the additive white generalized Gaussian noise (AWGGN) channel. The generalized Gaussian distribution is an effective model to describe real-world systems with impulsive noise. It can be specialized to the Laplace and Gaussian distributions depending on the choices of the decay rates of the heavy tail. By deriving the optimal IS distribution on the one-dimensional space mapped from the observation space, we present a general framework for designing IS estimators for linear block codes over a memoryless continuous channel. We apply the framework to the AWGGN channel and propose a noise-magnitude-domain IS (NM-IS) estimator. As an efficiency measure, the asymptotic IS gain of the proposed estimator is derived in a multiple integral as the signal-to-noise ratio (SNR) tends to infinity. Specifically, we reduce the gains of both Laplace and Gaussian noise cases to a one-dimensional integral. In addition, by limiting the use of the union bound to an L1-norm sphere with an optimized radius, we derive the sphere bound for the additive white Laplace noise channel. Simulation results show the efficiency of the proposed IS estimator in terms of the sample size and verify the accuracy of the prediction of the derived IS gain. Besides, we specialize the proposed framework to further improve the efficiency of the IS simulation based on the side information of the modulation. For the AWGGN channel with binary phase-shift keying (BPSK) modulation, we propose an enhanced NM-IS estimator, which outperforms the NM-IS estimator in efficiency. For the additive white Gaussian noise channel with M-PSK modulation, we propose an angular domain IS estimator and provide its efficiency analysis in terms of the asymptotic IS gain. Furthermore, in the performance evaluation of communication systems, it is often seen that researchers are more interested in the required SNR at a target error probability Pe than the other way around. The conventional solution to this Pe-targeted SNR estimation problem involves interpolation of simulated error probabilities at multiple handpicked SNRs for estimating the required SNR at the target error probability. We propose a Pe-targeted iterative IS estimator that solves the problem algorithmically. The proposed method can achieve the same efficiency (in terms of sample size) as estimating the error probability at the required SNR and allow the confidence interval of the SNR estimate to be obtained.