The problem of low complexity, close to optimal, channel decoding of linear codes with short to moderate block length is considered. It is shown that deep learning methods can be used to improve a standard belief propagation decoder, despite the large example space. Similar improvements are obtained for the min-sum algorithm. It is also shown that tying the parameters of the decoders across iterations, so as to form a recurrent neural network architecture, can be implemented with comparable results. The advantage is that significantly less parameters are required. We also introduce a recurrent neural decoder architecture based on the method of successive relaxation. Improvements over standard belief propagation are also observed on sparser Tanner graph representations of the codes. Furthermore, we demonstrate that the neural belief propagation decoder can be used to improve the performance, or alternatively reduce the computational complexity, of a close to optimal decoder of short BCH codes.

Abstract-A novel deep learning method for improving the belief propagation algorithm is proposed. The method generalizes the standard belief propagation algorithm by assigning weights to the edges of the Tanner graph. These edges are then trained using deep learning techniques. A well-known property of the belief propagation algorithm is the independence of the performance on the transmitted codeword. A crucial property of our new method is that our decoder preserved this property. Furthermore, this property allows us to learn only a single codeword instead of exponential number of codewords. Improvements over the belief propagation algorithm are demonstrated for various high density parity check codes.

We consider a sensor array located in an enclosure, where arbitrary transfer functions (TFs) relate the source signal and the sensors. The array is used for enhancing a signal contaminated by interference. Constrained minimum power adaptive beamforming, which has been suggested by Frost and, in particular, the generalized sidelobe canceler (GSC) version, which has been developed by Griffiths and Jim, are the most widely used beamforming techniques. These methods rely on the assumption that the received signals are simple delayed versions of the source signal. The good interference suppression attained under this assumption is severely impaired in complicated acoustic environments, where arbitrary TFs may be encountered. In this paper, we consider the arbitrary TF case. We propose a GSC solution, which is adapted to the general TF case. We derive a suboptimal algorithm that can be implemented by estimating the TFs ratios, instead of estimating the TFs. The TF ratios are estimated by exploiting the nonstationarity characteristics of the desired signal. The algorithm is applied to the problem of speech enhancement in a reverberating room. The discussion is supported by an experimental study using speech and noise signals recorded in an actual room acoustics environment.

We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random-coset analysis to produce an effect that is similar to output-symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q − 1 dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the AWGN channel, including results within 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz. Index TermsBandwidth efficient coding, coset codes, iterative decoding, low-density parity-check (LDPC) codes. not output-symmetric, thus posing a problem to their analysis. To overcome this problem, Hou et al. used coset LDPC codes rather than plain LDPC codes. The use of coset-LDPC codes was first suggested by Kavcić et al. [19] in the context of LDPC codes for channels with intersymbol interference (ISI).MLC and BICM codes are frequently decoded using multistage and parallel decoding, respectively. Both methods are suboptimal in comparison to methods that rely only on belief-propagation decoding 1 . Full belief-propagation decoding was considered by Varnica et al. [37] for MLC and by ourselves in [1] (using a variant of BICM LDPC called BQC-LDPC). However, both methods involve computations that are difficult to analyze.An alternative approach to designing nonbinary codes starts off with nonbinary LDPC codes. Gallager [16] defined arbitrary-alphabet LDPC codes using modulo-q arithmetic. Nonbinary LDPC codes were also considered by Davey and MacKay [10] in the context of codes for binary-input channels. Their definition uses Galois field (GF(q)) arithmetic. In this paper we focus on GF(q) LDPC codes similar to those suggested in [10].In [1] we considered coset GF(q) LDPC codes under maximum-likelihood (ML) decoding. We showed that appropriately designed coset GF(q) LDPC codes are capable of achieving the capacity of any discrete-memoryless channel. In this paper, we examine coset GF(q) LDPC codes under iterative decoding.A straightforward implementation of the nonbinary belief-propagation decoder has a very large decoding complexity. However, we discuss an implementation method suggested by Richardson and Urbanke [28][Section V] that us...

No abstract

We show how asymptotic estimates of powers of polynomials with non-negative coefficients can be used in the analysis of low-density parity-check (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ensembles. We then consider the binary erasure channel (BEC). Using these estimates we derive lower bounds on the error exponent, under iterative decoding, of LDPC codes used over the BEC. Both regular and irregular code structures are considered. These bounds are compared to the corresponding bounds when optimal (maximum likelihood) decoding is applied. Index Terms-Binary erasure channel (BEC), Code ensembles, Code spectrum, Iterative decoding, Low-density parity-check (LDPC) codes. I Introduction Various combinatorial problems can be solved using enumerating functions. These problems involve powers of polynomials with non-negative coefficients. For example, the spectrum of lowdensity parity-check (LDPC) codes [10] can often be expressed using an enumerating function. Another example is the analysis of the error probability of LDPC codes over the binary erasure channel (BEC) when iterative decoding is applied [6].

No abstract

Abstract-We propose an efficient maximum likelihood decoding algorithm for decoding low-density parity-check codes over the binary erasure channel. We also analyze the computational complexity of the proposed algorithm.