Linear Coding Algorithm Research Articles

The syndromes decoding algorithm in group codes

Our aim is the design of an efficient decoding algorithm in group codes. The algorithm is inspired by the well known syndrome decoding algorithm for linear codes and uses the decomposition of a semisimple group algebra KG as a direct sum of two-sided ideals, each of them generated by a central idempotent of KG. When G is an abelian group the algorithm can be modified to make it very simple and efficient. Some illustrative examples are presented.

Decoding of linear codes for single error bursts correction based on the determination of certain events

Introduction: In modern systems for communication, data storage and processing the error-correction capability of codes are estimated for memoryless channels. In real channels the noise is correlated, which leads to grouping error in bursts. A traditional method to fight this phenomenon is channel decorrelation, which does not allow developing of coding schemes, mostly utilizing the channel capacity. Thus the development of bursts decoding algorithms for arbitrary linear codes is the actual task. Purpose: To develop a single error burst decoding algorithm for linear codes, to estimate the decoding error probability and computational complexity. Results: Two approaches are proposed to burst error correction. The first one is based on combining the window sliding modification of well-known bit-flipping algorithm with preliminary analysis of the structure of parity check matrix. The second one is based on the recursive procedure of constructing the sequence of certain events which, in the worst case, performs the exhaustive search of error bursts, but in many cases the search may be significantly decreased by using the proposed heuristics. The proposed recursive decoding algorithm allows a guaranteed correction of any single error bursts within burst-correction capability of the code, and in many cases beyond the burst-correction capability. The complexity of this algorithm is significantly lower than that of a bit flipping algorithm if the parity-check matrix of the code is sparse enough. An alternative hybrid decoding algorithm is proposed utilizing the bit-flipping approach and showing the error probability and completion time comparable to the recursive algorithm, however, in this case the possibility of a guaranteed burst correction hardly can be proved. Practical relevance: The proposed decoding methods may be used in modern and perspective communication systems, allowing energy saving and increasing reliability of data transmission by better error performance and computational complexity.

Decoding Linear Codes over Chain Rings Given by Parity Check Matrices

We design a decoding algorithm for linear codes over finite chain rings given by their parity check matrices. It is assumed that decoding algorithms over the residue field are known at each degree of the adic decomposition.

A matrix based list decoding algorithm for linear codes over integer residue rings

In this paper we address the problem of list decoding of linear codes over an integer residue ring Zq, where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Zpr, called the standard form, and the p-adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Zpr. From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field pr−1Zpr. This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal.

On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings

Let p be a prime, s be a positive integer, and R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ε R, λ-constacyclic codes of length ps over R are ideals of the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> -λ). In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring. When R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring, all λ-constacyclic codes of length ps over R are known. In this paper, we establish the algebraic structures of all λ-constacyclic codes of length ps over R when R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and RT weight distributions of all constacyclic codes of length ps over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length ps over R is maximumdistance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length ps over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics.

Analysis of privacy and stability to interference in the communication channel of network steganography methods

In this paper, for the first time, the effectiveness of network steganography methods has been investigated using channel coding of data when they are transmitted over communication channels with interferences, and the method of mathematical modeling evaluates the stability of network steganography methods to the detection. The network steganography methods concealing data in such protocols as HTTP, TCP and ICMP were considered. Using the ICMP «Secure» mode of the developed method, the greatest throughput was achieved. To study the noise resistance of the methods before transmission by the communication channel, the packets were encoded using the 2 Binary 1 Quandary linear coding algorithm, after which additive white Gaussian noise was added to the stego object. The paper substantiates the possibility of using network steganography methods to protect biometric data from unauthorized access in the case of using remote biometric authentication. Based on the research results, it was determined that the network steganography method based on hiding data in TCP is more efficient when working via communication channels with noise. It makes possible to recover a message with small distortions at a dispersion value of 0.4. But according to a set of criteria for resistance to noise and detection probability, the best is the «Fast» mode of the network steganography method based on hiding data in ICMP.

Accurate target tracking via Gaussian sparsity and locality-constrained coding in heavy occlusion

This paper presents a Gaussian sparse representation cooperative model for tracking a target in heavy occlusion video sequences by combining sparse coding and locality-constrained linear coding algorithms. Different from the usual method of using l1-norm regularization term in the framework of particle filters to form the sparse collaborative appearance model (SCM), we employed the l1-norm and l2-norm to calculate feature selection, and then encoded the candidate samples to generate the sparse coefficients. Consequently, our method not only easily obtained sparse solutions but also reduced reconstruction error. Compared to state-of-the-art algorithms, our scheme achieved better performance in heavy occlusion video sequences for tracking a target. Extensive experiments on target tracking were carried out to show the results of our proposed algorithm compared with various other target tracking methods.

A Dual Locality-Constrained Linear Coding Algorithm for Image Classification

The locality-constrained linear coding (LLC) algorithm has been developed as an effective means for image classification. This algorithm uses locality linear constraints to encode feature points of the image, achieving higher classification accuracy than that of Spatial Pyramid Matching (SPM), spatial pyramid matching using sparse coding, and other traditional algorithms. However, the LLC algorithm uses only the locality information for the visual words in the dictionary while rarely using mutual information between the neighbouring feature points, causing severe ambiguities. Based on the principle of the locality correlation of images, we propose a new algorithm, named the dual locality-constrained linear coding algorithm (DLLC), which applies the locality information of the feature points and visual words and uses the discriminant information provided by the nearest neighbouring feature points. The experimental results demonstrate that the accuracy of the DLLC algorithm is higher than that of the LLC algorithm, particularly when the image category is large, but the set of dictionary training data is small.

Kernelized temporal locality learning for real-time visual tracking

Linear representation-based methods play an important role in the development of the target appearance modeling in visual tracking. However, such linear representation scheme cannot accurately depict the nonlinearly distributed appearance variations of the target, which often leads to unreliable tracking results. To fix this issue, we introduce the kernel method into the locality-constrained linear coding algorithm to comprehensively exploit its nonlinear representation ability. Further, to fully consider the temporal correlation between neighboring frames, we develop a point-to-set distance metric with L2, 1 norm as the temporal smoothness constraint, which aims to guarantee that the object between the two consecutive frames should be represented by the similar dictionaries temporally. Experimental results on Object Tracking Benchmark show that the proposed tracker achieves promising performance compared with other state-of-the-art methods.

Salient object detection via global and local cues

Previous saliency detection algorithms used to focus on low level features directly or utilize a bunch of sample images and manually labeled ground truth to train a high level learning model. In this paper, we propose a novel coding-based saliency measure by exploring both global and local cues for saliency computation. Firstly, we construct a bottom-up saliency map by considering global contrast information via low level features. Secondly, by using a locality-constrained linear coding algorithm, a top-down saliency map is formulated based on the reconstruction error. To better exploit the local and global information, we integrate the bottom-up and top-down maps as the final saliency map. Extensive experimental results on three large benchmark datasets demonstrate that the proposed approach outperforms 22 state-of-the-art methods in terms of three popular evaluation measures, i.e., the Precision and Recall curve, Area Under ROC Curve and F-measure value. Furthermore, the proposed coding-based method can be easily applied in other methods for significant improvement.

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

A binary nonlinear code can be represented as a union of cosets of a binary linear subcode. In this paper, the complexity of some algorithms to obtain this representation is analyzed. Moreover, some properties and constructions of new codes from given ones in terms of this representation are described. Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes. All results are written in such a way that they can be easily transformed into algorithms, and the performance of these algorithms is evaluated.

Parity-Check Matrices Separating Erasures From Errors

Most decoding algorithms of linear codes, in general, are designed to correct or detect errors. However, many channels cause erasures in addition to errors. In principle, decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code. For any given linear code and any given maximum number of correctable erasures, parity-check matrices are introduced that yield parity-check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. These matrices allow for the separation of erasures from errors to facilitate decoding. Several constructions of such separating parity-check matrices are presented. To reduce decoding complexity, separating parity-check matrices with small number of rows are preferred. The minimum number of rows in a parity-check matrix separating a given maximum number of erasures is called the separating redundancy. Upper and lower bounds on the separating redundancies are derived. In particular, it is shown that the separating redundancies tend to grow linearly with the number of rows in full-rank parity-check matrices of codes. The separating redundancies of some classes of codes are determined for some maximum numbers of erasures.

A Random Linear Coding Algorithm for Cognitive Wireless Mesh Networks

A Random Linear Coding Algorithm for Cognitive Wireless Mesh Networks

A search algorithm for linear codes: progressive dimension growth

This paper presents an algorithm, called progressive dimension growth (PDG), for the construction of linear codes with a pre-specified length and a minimum distance. A number of new linear codes over GF(5) that have been discovered via this algorithm are also presented.

Linear Codes over $\mathbb{F}_{q}[u]/(u^s)$ with Respect to the Rosenbloom–Tsfasman Metric

We investigate the structure of codes over $\mathbb{F}_q[u]/(u^s)$ rings with respect to the Rosenbloom-Tsfasman (RT) metric. We define a standard form generator matrix and show how we can determine the minimum distance of a code by taking advantage of its standard form. We define MDR (maximum distance rank) codes with respect to this metric and give the weights of the codewords of an MDR code. We explore the structure of cyclic codes over $\mathbb{F}_q[u]/(u^s)$ and show that all cyclic codes over $\mathbb{F}_q[u]/(u^s)$ rings are MDR. We propose a decoding algorithm for linear codes over these rings with respect to the RT metric.

Development of Walsh linear coding and its application to speech recognition

In this paper a Walsh linear coding algorithm (WLC) is developed which yields a compressed approximation to the Walsh power spectrum of a time frame. This algorithm is interpolative, rather than predictive. It minimizes the mean square interpolative error, and it yields WLC model spectra. The effectiveness of utilizing WLC model spectra in speech recognition systems is explored. Each of two unity gain distortion measures is embedded in a dynamic time warp based speaker-dependent isolated word recognition system and is tested by means of a 10-digit vocabulary.

Hard- and soft-decision decoding beyond the half minimum distance---An algorithm for linear codes (Corresp.)

A decoding algorithm for linear codes that uses the minimum weight words of the dual code as parity checks is defined. This algorithm is able to correct beyond the half minimum distance and has the capability of including soft-decision decoding. Results on applying this algorithm to quadratic residue (QR) codes, BCH codes, and the Golay codes (with and without soft-decision decoding) are presented.

Efficient decoding algorithms for linear codes on soft-quantized channels (Ph.D. Thesis abstr.)

Efficient decoding algorithms for linear codes on soft-quantized channels (Ph.D. Thesis abstr.)