Code Length Increases Research Articles

Blind Recognition of Reed-Solomon Codes Based on Galois Field Fourier Transform and Reliability Verification

The Galois field Fourier transform (GFFT) method has been widely applied in the blind recognition of Reed-Solomon (RS) codes in recent years. However, as the code length increases, the computational complexity increases rapidly. Also, blind recognition in a poor channel environment remains challenging. To solve the above challenges, a blind recognition algorithm is proposed based on GFFT and reliability verification (RV). Low-complexity GFFT (LC-GFFT) is proposed to calculate the shortened spectrums, reducing the computational complexity and improving the correct codeword utilization rate (CUR). Next, the criteria for parameter estimation (PE) are given. The RV module is designed to verify the correctness of the estimated parameters. The simulations demonstrate that our algorithm can recognize RS(15, 11, 2, 19) and RS(31, 15, 8, 37) with over 95% correct rate when SNR is above 0.8dB and 2.5dB, achieving at least 7dB of signal-to-noise (SNR) defined gain over the existing algorithms. Moreover, the recognition time of our method is reduced by 74.8% compared to that of the GFFT-based method for RS(31, 15, 8, 37) when <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 1 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> .

A Revisit of Hashing Algorithms for Approximate Nearest Neighbor Search

Approximate Nearest Neighbor Search (ANNS) is a fundamental problem in many areas of machine learning and data mining. During the past decade, numerous hashing algorithms are proposed to solve this problem. Every proposed algorithm claims to outperform Locality Sensitive Hashing (LSH), which is the most popular hashing method. However, the evaluation of these hashing article was not thorough enough, and the claim should be re-examined. If implemented correctly, almost all the hashing methods will have their performance improved as the code length increases. However, many existing hashing article only report the performance with the code length shorter than 128. In this article, we carefully revisit the problem of search-with-a-hash-index and analyze the pros and cons of two popular hash index search procedures. Then we proposed a simple but effective novel hash index search approach and made a thorough comparison of eleven popular hashing algorithms. Surprisingly, the random-projection-based Locality Sensitive Hashing ranked the first, which is in contradiction to the claims in all the other 10 hashing article. Despite the extreme simplicity of random-projection-based LSH, our results show that the capability of this algorithm has been far underestimated. For the sake of reproducibility, all the codes used in the article are released on GitHub, which can be used as a testing platform for a fair comparison between various hashing algorithms.

Capacity-Approaching Polar Codes With Long Codewords and Successive Cancellation Decoding Based on Improved Gaussian Approximation

This paper focuses on an improved Gaussian approximation (GA) based construction of polar codes with successive cancellation (SC) decoding over an additive white Gaussian noise (AWGN) channel. Arıkan proved that polar codes with low-complexity SC decoding can approach the channel capacity of an arbitrary symmetric binary-input discrete memoryless channel, provided that the code length is chosen large enough. Nevertheless, how to construct such codes over an AWGN channel with low computational effort has been an open problem. Compared to density evolution, the GA is known as a low complexity yet powerful technique that traces the evolution of the mean log likelihood ratio (LLR) value by iterating a nonlinear function. Therefore, its high-precision numerical evaluation is critical as the code length increases. In this work, by analyzing the asymptotic behavior of this nonlinear function, we propose an improved GA approach that makes an accurate trace of mean LLR evolution feasible. With this improved GA, through numerical analysis and simulations with code lengths up to N = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">18</sup> , we explicitly demonstrate that various code-rate polar codes with long codeword and capacity approaching behavior can be easily designed.

Polar codes Bhattacharyya parameter generalisation

Recently, polar codes were proposed by Arikan to achieve optimum channel capacity given by Shannon theorem with low encoding and decoding complexity. Polar code construction depends on two main foundation criteria which are kernel matrix and Bhattacharyya parameter. They are related to each other, therefore the selection method for both affects the performance of polar code. Firstly, in this study, the derivations of the bounds for Bhattacharyya parameter are proved and generalised together with a proposed method to select the best kernel matrix to achieve the optimum capacity. Then, recursive channel transformations and successive cancellation decoding of the selected 3 × 3 best kernel matrix are proved. Furthermore, a general formula for polar code complexity and hardware implementation has been discussed. Simulation results show that the achievable bit error rate for the proposed methodology of selection is the same as some existing methods with the same order of complexity, which indicates its effectiveness. Polar code performance for the selected 3 × 3 best kernel matrix is improved as the code length increases. Moreover, this proposed method is general to be for higher dimension kernel matrices.

Fast Encoding Algorithms for Reed–Solomon Codes With Between Four and Seven Parity Symbols

This article describes a fast Reed–Solomon encoding algorithm with four and seven parity symbols in between. First, we show that the syndrome of Reed–Solomon codes can be computed via the Reed–Muller transform. Based on this result, the fast encoding algorithm is then derived. Analysis shows that the proposed approach asymptotically requires 3 XORs per data bit, representing an improvement over previous algorithms. The simulation demonstrates that the performance of the proposed approach improves with the increase of code length and is superior to other methods. In particular, when the parity number is 5, the proposed approach is about two times faster than other cutting-edge methods.

The Construction of Permutation Group Codes for Communication Systems: Prime or Prime Power?

This paper focuses on correcting a theorem relating to the construction of permutation group codes (PGCs). The theorem in question assumed that affine transformation could be used to enumerate all the code words of a permutation array with a minimum Hamming distance (MHD) of n - 1 for any n > 1. This assumption was founded upon the proposition that, if the code length, n, is a prime power, then the maximum cardinality of the code will be n(n - 1) and its MHD will be n - 1. However, two typical algebraic methods, one relying on affine transformation and another upon the composite operation of two small subgroups of a symmetric group, can violate this proposition. This is because it is only when n is a prime rather than a prime power that they can enumerate all the code words of an (n; n (n - 1) ; n - 1)-PGC. By investigating how the range of n impacts upon the cardinality and MHD of a code, we provide a corrective theorem that stipulates the construction of (p q ; p 2q (1 - 1/p); p q (1 - 1/p))-PGCs when n = pq is a prime power, wherep is a prime and q > 1. On the basis of this theorem, and under the condition of n being the power of 2, we construct a (2 q ; 2 2q-1 ; 2 q-1 )-PGC and present an encoder that can map a k-bit binary information sequence to an n = 2 2q -dimension permutation code word. The natural array structure of (2 q ; 2 2q-1 ; 2 q-1 )-PGCs makes them especially well-suited to forming the basis of low-complexity encoders. We present simulation-based experiments that show that, as the code length increases, the performance of these codes improves. The best performance is for a (16; 128; 8)-PGC, which can achieve -3.8 dB with a word error rate of 10 -7 .

The Generation of $(n,n(n-1),n-1)$ Permutation Group Codes for Communication Systems

This paper concerns the construction and hardware execution of (n, n(n-1), n-1) permutation group codes (PGCs) for communication systems. Cyclic shift techniques acting on a permutation, including cyclic shift operators and their compound functions (CFs), are defined and demonstrated, then compared to other tools such as cyclic subgroups and n-dimension cyclic shift registers. A generalized finite state machine is established that can enumerate n! permutations based on complete CFs. This is a hardware-oriented algorithm that is capable of being significantly faster than all existing algorithms which we can find up to now. After this, the ideal structural features for an ordered set of n! permutations are discussed and the multiset features of (n, n(n-1), n-1)-PGCs in this ordered set are pinpointed. A simpler way of generating (n, n(n-1), n-1)-PGCs is then developed using cyclic shift techniques, where the code length n is a prime. This outperforms existing approaches using composite operation and affine transform on module-n operation. The simulation experiments in AWGN channel show that the performance on codeword error rates (WER) and signal-noise ratio (SNR) becomes better as the code length increases and the best SNR performance achieves −0.8 dB at 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> WER for code length n=11.

Reduced Complexity Window Decoding of Spatially Coupled LDPC Codes for Magnetic Recording Systems

In channel coding theory, the performance of error correcting codes (ECCs) approaching the Shannon limit can be achieved through increasing code lengths. Unfortunately, the complexity of ECCs will be increased as the code length increases. Nowadays, the magnetic recording (MR) system takes advantage of powerful ECCs by using 4 Kbytes sector. Among the advanced ECCs, the spatially coupled LDPC (SC-LDPC) codes (also known as a LDPC convolutional code) [1]are shown to have the decoding latency and complexity lower than those of the underlying LDPC block codes (LDPC-BC). Moreover, the SC-LDPC codes with threshold decoding outperform the LDPC-BC codes [2]. Hence, the SC-LDPC codes are the strong candidate for the future MR systems, when the sector size is increased beyond 4 Kbytes. An SC-LDPC decoder can use sliding window decoding [3]whereby the received signals are decoded by sliding window along the bit sequence. The window decoder is called “uniform window decoding (U-WD),” when all variable nodes (VNs) within a window are updated. In order to reduce the complexity of window decoding, some researchers proposed the non-uniform window decoding (N-WD) [4], which do not update the VNs with no improvement in the bit error rate (BER). This approach provides about 35-50% reduction in complexity compared to U-WD. In this work, we consider the application of SC-LDPC codes in MR systems, whereby SC-LDPC decoder cooperates with BCJR detector to encounter inter-symbol interference (ISI). We propose the dynamic shifting of window decoding (DS-WD) to reduce the complexity of SC-LDPC codes. Herein, the number of shifted bits is defined according to their soft BERs which are estimated at each decoding position. In addition, we modify the N-WD [4]to reinforce our proposed algorithm called “dynamic-shifting non-uniform window decoding (DS-N-WD).” The DS-WD and DS-N-WD achieve the complexity reduction of 7% and 25% without any loss in performance compared to the N-WD algorithms.

Moderate Deviation Analysis for Classical Communication over Quantum Channels

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altug and Wagner as well as Polyanskiy and Verdu. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

On Joint Source-Channel Coding for a Multivariate Gaussian on a Gaussian MAC

In this paper, nonlinear distributed joint source-channel coding (JSCC) schemes for transmission of multivariate Gaussian sources over a Gaussian multiple access channel are proposed and analyzed. The main contribution is a zero-delay JSCC named Distributed Quantizer Linear Coder (DQLC), which performs relatively close the information theoretical bounds, improves when the correlation among the sources increases, and does not level off as the signal-to-noise ratio (SNR) becomes large. Therefore it outperforms any linear solution for sufficiently large SNR. Further an extension of DQLC to an arbitrary code length named Vector Quantizer Linear Coder (VQLC) is analyzed. The VQLC closes in on the performance upper bound as the code length increases and can potentially achieve the bound for any number of independent sources. The VQLC leaves a gap to the bound whenever the sources are correlated, however. JSCC achieving the bound for arbitrary correlation has been found for the bivariate case, but that solution is significantly outperformed by the DQLC/VQLC when there is a low delay constraint. This indicates that different approaches are needed to perform close to the bounds when the code length is high and low. The VQLC/DQLC also apply for bandwidth compression of a multivariate Gaussian transmitted on point-to-point links.

A Novel Encoding Scheme for Effective Biometric Discretization: Linearly Separable Subcode

Separability in a code is crucial in guaranteeing a decent Hamming-distance separation among the codewords. In multibit biometric discretization where a code is used for quantization-intervals labeling, separability is necessary for preserving distance dissimilarity when feature components are mapped from a discrete space to a Hamming space. In this paper, we examine separability of Binary Reflected Gray Code (BRGC) encoding and reveal its inadequacy in tackling interclass variation during the discrete-to-binary mapping, leading to a tradeoff between classification performance and entropy of binary output. To overcome this drawback, we put forward two encoding schemes exhibiting full-ideal and near-ideal separability capabilities, known as Linearly Separable Subcode (LSSC) and Partially Linearly Separable Subcode (PLSSC), respectively. These encoding schemes convert the conventional entropy-performance tradeoff into an entropy-redundancy tradeoff in the increase of code length. Extensive experimental results vindicate the superiority of our schemes over the existing encoding schemes in discretization performance. This opens up possibilities of achieving much greater classification performance with high output entropy.

Hardware Implementation of Successive-Cancellation Decoders for Polar Codes

The recently-discovered polar codes are seen as a major breakthrough in coding theory; they provably achieve the theoretical capacity of discrete memoryless channels using the low-complexity successive cancellation decoding algorithm. Motivated by recent developments in polar coding theory, we propose a family of efficient hardware implementations for successive cancellation (SC) polar decoders. We show that such decoders can be implemented with O(N) processing elements and O(N) memory elements. Furthermore, we show that SC decoding can be implemented in the logarithmic domain, thereby eliminating costly multiplication and division operations, and reducing the complexity of each processing element greatly. We also present a detailed architecture for an SC decoder and provide logic synthesis results confirming the linear complexity growth of the decoder as the code length increases.

Quantum Serial Turbo Codes

In this paper, we present a theory of quantum serial turbo codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum low-density parity-check (LDPC) codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Second, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular, its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoders have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be noncatastrophic and nonrecursive. While the resulting families of turbo codes have bounded minimum distance, from a pragmatic point of view, the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well-chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.

A One-to-One Code and Its Anti-Redundancy

One-to-one codes are ldquoone-shotrdquo codes that assign a distinct codeword to source symbols and are not necessarily prefix codes (more generally, uniquely decodable). Interestingly, as Wyner proved in 1972, for such codes the average code length can be smaller than the source entropy. By how much? We call this difference the anti-redundancy. Various authors over the years have shown that the anti-redundancy can be as big as minus the logarithm of the source entropy. However, to the best of our knowledge precise estimates do not exist. In this note, we consider a block code of length n generated for a binary memoryless source, and prove that the average anti-redundancy is -1/2 log2 n +C +F (n)+o (1) where C is a constant and either F (n) = 0 if log2(1-p)/p is irrational (where p is the probability of generating a ldquo0rdquo) or F(n) is a fluctuating function as the code length increases. This relatively simple finding requires a combination of analytic tools such as precise evaluation of Bernoulli sums, the saddle point method, and theory of distribution of sequences modulo 1.

A new family of two-dimensional codes for optical CDMA systems

The design of a new family of two-dimensional single pulse per column codes for optical code division multiple access (OCDMA) networks is reported. The 1-D modified pseudo-noise codes have been known to be orthogonal and their generation and system design based on these codes is rather simple. But their performance is limited due to the bandwidth constraints if the code length increases. Hence, using these 1-D modified pseudo-noise codes, modified 2-D pseudo-noise matrix codes (MPMCs) are generated. The system performance is evaluated for two, three and four simultaneous users using the link with all the sources responsible for degradation included: attenuation, chromatic dispersion, non-linear refractive effects, non-linear scattering and four-wave mixing. The effect of the non-linear and lossy dispersive medium over the system performance is shown by plotting the BER with respect to the link length for the systems designed using encoders/decoders base on 1-D modified pseudo-noise codes and our MPMCs. The performance is compared for the two types of codes by finding the crosstalk due to interfering users simultaneously operating in the network.

Performance characteristics and weight distribution analysis of turbo product code with Reed–Muller component codes

In this paper, we present simulation results for Reed–Muller (RM) turbo codes over AWGN and Rayleigh fading channels. We also compare the simulation results of turbo product codes and block turbo codes with RM component codes over an AWGN channel. We show that minimum distance is not important as far as the BER performance of long codes is concerned. The weight distribution of RM codes of different lengths and turbo product codes with first-order RM component codes are obtained and analyzed for their good performance. We show that the weight distribution asymptotically approaches that of random coding as the code length increases.

Source-optimized irregular repeat accumulate codes with inherent unequal error protection capabilities and their application to scalable image transmission

The common practice for achieving unequal error protection (UEP) in scalable multimedia communication systems is to design rate-compatible punctured channel codes before computing the UEP rate assignments. This paper proposes a new approach to designing powerful irregular repeat accumulate (IRA) codes that are optimized for the multimedia source and to exploiting the inherent irregularity in IRA codes for UEP. Using the end-to-end distortion due to the first error bit in channel decoding as the cost function, which is readily given by the operational distortion-rate function of embedded source codes, we incorporate this cost function into the channel code design process via density evolution and obtain IRA codes that minimize the average cost function instead of the usual probability of error. Because the resulting IRA codes have inherent UEP capabilities due to irregularity, the new IRA code design effectively integrates channel code optimization and UEP rate assignments, resulting in source-optimized channel coding or joint source-channel coding. We simulate our source-optimized IRA codes for transporting SPIHT-coded images over a binary symmetric channel with crossover probability p. When p = 0.03 and the channel code length is long (e.g., with one codeword for the whole 512 x 512 image), we are able to operate at only 9.38% away from the channel capacity with code length 132380 bits, achieving the best published results in terms of average peak signal-to-noise ratio (PSNR). Compared to conventional IRA code design (that minimizes the probability of error) with the same code rate, the performance gain in average PSNR from using our proposed source-optimized IRA code design is 0.8759 dB when p = 0.1 and the code length is 12800 bits. As predicted by Shannon's separation principle, we observe that this performance gain diminishes as the code length increases.

Design of multilength optical orthogonal codes for optical CDMA multimedia networks

To simultaneously support multimedia services with different signaling rates and quality-of-service requirements in optical code division multiple access (CDMA) networks, a new class of multilength, constant-weight optical orthogonal codes (OOCs) with good correlation properties is constructed algebraically in this paper. The performance of these new OOCs in an optical CDMA system with double-media services is analyzed. In contrast to conventional CDMA, our study shows that the performance of these multilength OOCs worsens as the code length increases, allowing prioritization in optical CDMA. Finally, an application of these multilength OOCs to integrate different types of multimedia services is briefly discussed.

Compile and Run-Time Support for the Parallelization of Sparse Matrix Updating Algorithms

This work presents a survey of the capabilities that the sparse computation offers for improving performance when parallelized, either automatically or through a data-parallel compiler. The characterization of a sparse code gets more complicated as code length increases: Access patterns change from loop to loop, thus making necessary to redefine the parallelization strategy. While dense computation solely offers the possibility of redistributing data structures, several other factors influence the performance of a code excerpt in the sparse field, like source data representation on file, compressed data storage in memory, the creation of new nonzeroes at run-time (fill-in) or the number of processors available. We analize the alternatives that arise from each issue, providing a guideline for the underlying compilation work and illustrating our techniques with examples on the Cray T3E.

Code synchronisation in two-path channel: effectof interpath interference on acquisition

An analysis and simulation results are presented of a DS spread spectrum code acquisition in a multipath channel which causes interpath interference (IPI). Noncoherent matched filter (MF) acquisition is used. Results can be applied also for acquisition methods based on correlation measurements using a stepped serial search. The results indicate that the influence of IPI is significant when the spreading codes are short but the effect decreases rapidly as the code length increases.