A bounded-distance decoding algorithm for lattices obtained from a generalized code formula

In this paper, we propose a near maximum likelihood (ML) decoding technique, which reduces the computational complexity of the exact ML decoding algorithm. The computations needed for the tree search in the ML decoding is simplified by reducing the dimension of the search space prior to the tree search. In order to compensate performance loss due to the dimension reduction, a list stack algorithm (LSA) is considered, which produces a list of the top K closest points. The combination of both approaches, called reduced dimension list stack algorithm (RD-LSA), is shown to provide flexibility and offers a performance-complexity trade-off. Simulations performed for V-BLAST transmission demonstrate that significant complexity reduction can be achieved compared to the sphere decoding algorithm (SDA) while keeping the performance loss below an acceptable level.

Maximum-likelihood (ML) decoding algorithms for Gaussian multiple-input multiple-output (MIMO) linear channels are considered. Linearity over the field of real numbers facilitates the design of ML decoders using number-theoretic tools for searching the closest lattice point. These decoders are collectively referred to as sphere decoders in the literature. In this paper, a fresh look at this class of decoding algorithms is taken. In particular, two novel algorithms are developed. The first algorithm is inspired by the Pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the Viterbo-Boutros sphere decoder. The connection between the proposed algorithm and the stack sequential decoding algorithm is then established. This connection is utilized to construct the second algorithm which can also be viewed as an application of the Schnorr-Euchner strategy to ML decoding. Aided with a detailed study of preprocessing algorithms, a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a near-ML detection performance. This claim is supported by intuitive arguments and simulation results in many relevant scenarios.

Let \(\mathcal R_{t,n}\) denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller code RM(2,6) in the sets \(\mathcal R_{t,6}\), t=0,1,2 is 16. Second, we show that the covering radius of the binary Reed-Muller code RM(2,7) in the set \(\mathcal R_{3,7}\) is 32. We derive a new lower bound for the covering radius of the Reed-Muller code RM(2,n) in the set \(\mathcal R_{n-1,4}\). Finally, we present new lower bounds in the sets \(\mathcal R_{t,7}\), t=0,1,2.

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

This paper introduces multidimensional generalizations of binary Reed-Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary Reed-Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multiple-input-multiple-output (MIMO) channel at both low and moderate signal-to-noise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings.

This letter considers the average complexity of maximum-likelihood (ML) decoding of convolutional codes. ML decoding can be modeled as finding the most probable path taken through a Markov graph. Integrated with the Viterbi algorithm (VA), complexity reduction methods often use the sum log likelihood (SLL) of a Markov path as a bound to disprove the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this letter, it is shown that SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols. It is demonstrated that such neighboring log likelihood (NLL)-based optimality tests, whose efficiency does not depend on the codeword length, can bring significant complexity reduction. The results are generalized to ML sequence detection in a class of discrete-time hidden Markov systems.

In this paper, Sphere Decoding (SD) algorithms for Spatial Modulation (SM) are developed to reduce the computational complexity of Maximum-Likelihood (ML) detectors. Two SDs specifically designed for SM are proposed and analysed in terms of Bit Error Ratio (BER) and computational complexity. Using Monte Carlo simulations and mathematical analysis, it is shown that by carefully choosing the initial radius the proposed sphere decoder algorithms offer the same BER as ML detection, with a significant reduction in the computational complexity. A tight closed form expression for the BER performance of SM-SD is derived in the paper, along with an algorithm for choosing the initial radius which provides near to optimum performance. Also, it is shown that none of the proposed SDs are always superior to the others, but the best SD to use depends on the target spectral efficiency. The computational complexity trade-off offered by the proposed solutions is studied via analysis and simulation, and is shown to validate our findings. Finally, the performance of SM-SDs are compared to Spatial Multiplexing (SMX) applying ML decoder and applying SD. It is shown that for the same spectral efficiency, SM-SD offers up to 84% reduction in complexity compared to SMX-SD, with up to 1 dB better BER performance than SMX-ML decoder.

Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding, possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is $1 - 3 delta$ for nonbinary codes, where an adversary is allowed to corrupt up to delta fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff, and it is below 1 - 3 delta. We show that achieving slightly larger correctness (as a function of $delta$) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our results hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for q-query codes for q>3, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms. In addition, our results imply tradeoffs on the parameters of error correcting data structures.

Transmission efficiency in radio channels can be considerably improved by using multiple transmit and receive antennas and employing a family of schemes called space-time (ST) coding. Both extended range and/or improved bandwidth efficiency can be achieved, compared with a radio link with a single transmit and receive antenna. Bit-interleaved coded modulation schemes give diversity gains on fading channels with higher order modulation constellations combined with conventional binary convolutional codes also for the case of a single transmit and receive antenna radio link. In this paper, we study a family of flexible bandwidth-efficient ST coding schemes which combine these two ideas in a narrowband flat-fading channel and single-carrier modems. We address receiver complexity for the case of a large number of transmit antennas and higher order modulation constellations. Especially, we focus on practical configurations, where the number of transmit antennas is greater than that of receive antennas. Simplified receivers using tentative decisions are proposed and evaluated by means of simulations. Tradeoffs between complexity reduction and performance loss are presented. We emphasize systems that are of particular interest in applications where the number of transmit antennas exceeds the number of receive antennas. A system with four transmit antennas with an eight-fold complexity reduction and a performance loss of about 1 dB is demonstrated

Locality-sensitive hashing (LSH) is a popular data-independent indexing method for approximate similarity search, where random projections followed by quantization hash the points from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. Most of high-dimensional visual descriptors for images exhibit a natural matrix structure. When visual descriptors are represented by high-dimensional feature vectors and long binary codes are assigned, a random projection matrix requires expensive complexities in both space and time. In this paper we analyze a bilinear random projection method where feature matrices are transformed to binary codes by two smaller random projection matrices. We base our theoretical analysis on extending Raginsky and Lazebnik's result where random Fourier features are composed with random binary quantizers to form locality sensitive binary codes. To this end, we answer the following two questions: (1) whether a bilinear random projection also yields similarity-preserving binary codes; (2) whether a bilinear random projection yields performance gain or loss, compared to a large linear projection. Regarding the first question, we present upper and lower bounds on the expected Hamming distance between binary codes produced by bilinear random projections. In regards to the second question, we analyze the upper and lower bounds on covariance between two bits of binary codes, showing that the correlation between two bits is small. Numerical experiments on MNIST and Flickr45K datasets confirm the validity of our method.

In cluster-based multihop networks, the Ad Hoc Routing (AHR) scheme has been proposed to reduce the implementation complexity of the optimal routing scheme. However, this complexity reduction also causes performance loss. In addition, when the practical distributed relay selection is considered, further performance loss occurs due to the relay selection error. In this paper, we exploit cooperative diversity to compensate these performance losses, address the problem of the conventional distributed relay selection scheme, and then propose a novel Distributed Ad Hoc Cooperative Routing (DAHCR) scheme. In the proposed approach, the sender selects not only the receiver but also one relay to form a cooperative communication to reduce the power consumption. Simulation results reveal that the performance improvements achieved by the cooperative diversity and the proposed DAHCR scheme are the trade-offs of increased complexity.

Spatially coupled low-density parity-check (SC-LDPC) codes have emerged to possess capacity-approaching performance. The SC-LDPC codes can be decoded by a sliding window, therefore, the decoding latency and complexity of SC-LDPC codes are lower than those of underlying LDPC codes when the codeword length is very large. In this paper, the SC-LDPC decoder with the sliding window is employed in turbo equalization of magnetic recording systems. We examine the bit error rates (BERs) of the output of the sliding window during the iterative decoding, and then observe that the consecutive code blocks have approximately the same BERs. Herein, to reduce decoding complexity of SC-LDPC codes, the consecutive code blocks can be considered as the output of SC-LDPC codes. In addition, the non-uniform schedule update is adopted in the window decoding to avoid unnecessary updates within a window. The simulation results show that the proposed decoding algorithms applied in bit-patterned media magnetic recording systems can achieve a significant reduction in complexity compared to the traditional decoding without any loss in BER performance.

The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and output-symmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the additive white Gaussian noise (AWGN) channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds

In this paper, we propose two complexity reduction techniques for blind adaptive beamforming in orthogonal frequency division multiplexing (OFDM) antenna array systems. For each sub-carrier, a beamformer with generalized sidelobe canceller (GSC) structure is designed according to the constrained constant modulus (CCM) criterion with weight vectors adaptation based on the recursive least squares (RLS) method. The two techniques that we propose for complexity reduction rely on frequency domain interpolation and spatial domain clustering, respectively. The former exploits the coherence bandwidth of the radio channel, so that only the weight vectors at selected frequencies need to be adapted, while the remaining weight vectors are obtained via interpolation. The latter relies on the partitioning of the receiving array into sub-arrays of smaller size. In addition, a combination of these two techniques is proposed to further reduce the complexity. A complexity analysis is presented to quantify the computational savings offered by the proposed techniques. Finally, simulation results are provided to illustrate that these savings can be obtained at the price of only a minor, acceptable loss in performance when compared to the direct application of the conventional CCM-RLS-GSC beamforming algorithm to the wideband OFDM antenna array systems.

The invention of Fountain codes is a major advance in the field of error correcting codes. The goal of this work is to study and develop algorithms for source and channel coding using a family of Fountain codes known as Raptor codes. From an asymptotic point of view, the best currently known sum-product decoding algorithm for non binary alphabets has a high complexity that limits its use in practice. For binary channels, sum-product decoding algorithms have been extensively studied and are known to perform well. In the first part of this work, we develop a decoding algorithm for binary codes on non-binary channels based on a combination of sum-product and maximum-likelihood decoding. We apply this algorithm to Raptor codes on both symmetric and non-symmetric channels. Our algorithm shows the best performance in terms of complexity and error rate per symbol for blocks of finite length for symmetric channels. Then, we examine the performance of Raptor codes under sum-product decoding when the transmission is taking place on piecewise stationary memoryless channels and on channels with memory corrupted by noise. We develop algorithms for joint estimation and detection while simultaneously employing expectation maximization to estimate the noise, and sum-product algorithm to correct errors. We also develop a hard decision algorithm for Raptor codes on piecewise stationary memoryless channels. Finally, we generalize our joint LT estimation-decoding algorithms for Markov-modulated channels. In the third part of this work, we develop compression algorithms using Raptor codes. More specifically we introduce a lossless text compression algorithm, obtaining in this way competitive results compared to the existing classical approaches. Moreover, we propose distributed source coding algorithms based on the paradigm proposed by Slepian and Wolf.