Permutation Matrices Research Articles

Поиск информационных совокупностей при исправлении пакетов ошибок квазициклическими кодами

This article discusses the question of assessing the probability of finding information sets in block-permutation and block-circulant matrices. Traditionally, interference-resistant coding is considered independent errors, however, in real systems they can be grouped and generate a so-called error burst. Known estimates of the probability of finding information sets are conducted for random matrices, and for correcting error bursts widespread block-permutation low density parity check codes (LDPC-codes) or block-circulant quasi cyclic codes (QC-codes) can be used. To estimate the probability of finding information sets mathematical modeling was used. Experiments have been carried out to identify parameters for specific structures that give the greatest probability of finding information sets. The article presents the results reflecting certain features in the values of the probability of finding information sets for matrices of different types, given assumptions and hypotheses about the features. Dependence of the presence of an information set from the size and location of its search interval inside the block permutation matrix was identified. The results of this research may be used to reduce the complexity of decoding by information sets, which, when considering random matrices, is exponential.

A New Design of Substitution Box with Ideal Strict Avalanche Criterion

The use of S-boxes (substitution boxes) to provide nonlinear properties is known to be a common way to design a block cipher. These nonlinear properties are necessary to ensure the security of a block cipher. This manuscript proposes a design construction of a new S-box using affine transformation via cellular automata as a permutation matrix. We incorporate this cellular-automaton permutation matrix into the AES S box structure and test various irreducible polynomials. Nonlinearity, bijection, bit independence criterion, strict avalanche effect, linear approximation probability, and differential uniformity are the standard performance requirements used to evaluate the S-boxes that arise. Using this method, we are able to determine an irreducible polynomial that enables the construction of a new S-box design that can achieve an ideal strict avalanche criterion (SAC), which will subsequently provide efficiency in the design of block ciphers.

Doubly stochastic and permutation solutions to AXA = XAX when A is a permutation matrix

Let A be a permutation matrix. We prove that the equation AXA=XAX has nontrivial doubly stochastic solutions if and only if A has at least one nonzero entry on its main diagonal. We provide an algorithm for obtaining some of these doubly stochastic solutions. Afterwards, we completely characterize the set of permutation solutions for this equation: we obtain the if and only if criteria for their existence, provide an algorithm for the exact solutions in closed form, prove that they are all non-commuting solutions for the initial equation and we calculate the cardinality of this set. These findings generalize some previously known results regarding this topic.

Minimum upper bound of the Cayley transform of an orthogonal matrix multiplied by signed permutation matrices

O'Dorney has proved that for any orthogonal matrix U, there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which |sij|≤1, where S=(sij)=Q(UD) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2−1 by multiplying by a certain signed permutation matrix, instead of a signature matrix.

Doubly Stochastic Matrices and Schur-Weyl Duality for Partition Algebras

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the linear span of that set. Thanks to a known Schur-Weyl duality, this gives a new basis for the centralizer algebra of the partition algebra acting on the $r$th tensor power of a vector space. We give some related results on the set of doubly stochastic matrices in that algebra.

Fusion steady‐state robust filtering for uncertain multisensor networked systems with application to autoregressive moving average signal estimates

AbstractThis article is concerned with the centralized fusion (CF) robust steady‐state Kalman filtering problem for a class of multisensor networked systems with mixed uncertainties including state‐dependent and noise‐dependent multiplicative noises, missing measurements, packet dropouts, and uncertain noise variances. By using a model transformation approach, the initial multisensor system under study is transformed into a multi‐model multisensor system only with uncertain noise variances. By introducing an augmented state vector, the CF system is obtained. In the light of the minimax robust estimation method, and based on the worst‐case fusion system with conservative upper bounds of uncertain noise variances, the CF robust steady‐state Kalman estimators (predictor, filter, and smoother) are proposed in a unified form. By means of the permutation matrices and Lyapunov equation method, the robustness of CF estimators is proved. The accuracy relations among the robust local and fusion steady‐state estimators are proved. An example applied to autoregressive moving average signal estimation is put forward, and the robust CF steady‐state signal estimators are proposed. Simulation experiment shows the correctness of the proposed approach.

Time domain speech scrambler based on particle swarm optimization

Abstract Speech scrambler is used to transform clear speech into an unintelligible signal to prevent eavesdropping. The speech scrambling algorithm involves the permutation of speech segments depending on a specific permutation matrix which may be fixed or dynamic during encryption. A fixed permutation matrix is easy to break and has given high residual intelligibility in the scrambled signal. The proposed scheme used a particle swarm optimization algorithm to generate a dynamic permutation array that can attain a high degree of security. The outcome of the scrambled speech signal does not have any residual intelligibility, and the quality of the descrambled speech is extremely satisfying, with zero mean squared error.

Constructing Grith Eight GC-LDPC Codes Based on the GCD FLRM Matrix with a New Lower Bound.

By connecting multiple short, local low-density parity-check (LDPC) codes with a global parity check, the globally coupled (GC) LDPC code can attain high performances with low complexities. The typical design of a local code is a quasi-cyclic (QC) LDPC for which the code length is proportional to the size of circulant permutation matrix (CPM). The greatest common divisor (GCD)-based full-length row multiplier (FLRM) matrix is constrained by a lower bound of CPM size to avoid six length cycles. In this paper, we find a new lower bound for the CPM size and propose an algorithm to determine the minimum CPM size and the corresponding FLRM matrix. Based on the new lower bound, two methods are proposed to construct the GC-QC-LDPC code of grith 8 based on the GCD based FLRM matrix. With the proposed algorithm, the CPM size can be 45% less than that given by sufficient condition of girth 8. Compared with the conventional GC-LDPC construction, the codes constructed with the proposed method have improved performance and are more flexible in code length and code rate design.

A characterization of orthogonal permutative matrices of order 4

Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and rational orthogonal permutative matrices of order 4. We show that any such matrix can always be expressed as a linear combination of up to four permutation matrices. Finally we determine several matrix spaces generated by linearly independent permutation matrices such that any orthogonal matrix in these spaces is always permutative or direct sum of orthogonal permutative matrices up to permutation of its rows and columns.

Performance analysis of optical spectral amplitude CDMA system unipolar codes with double weight construction design

Abstract Optical Codes Division Multiple Access is a multiplexing approach in which the transmitter a distinguishable optical code is assigned to each user in the communication channel. OCDMA is designed for radio frequency (RF) communication systems and it is a variation of the CDMA system. Subsequently, to develop all OCDMA communication systems, the success of CDMA in wireless networks, as well as the full benefits of fibre optics, such as good security, extremely high bandwidth, low attenuation, and long-haul transmission. In this research Spectral Amplitude Coding-Optical Code Division Multiple Access is proposed. Initially, Modified Double Weight and Zero Cross-Correlation (MDW-ZCC) is proposed in this research to construct the code design of SAC-OCDMA, it produces the best possible autocorrelation and cross-correlation values. Subsequently, a combined Single Photodiode (SPD) and Direct detection technique is introduced to decode the transmitter information and it reduces the noise. Accordingly, a Gaussian approximation technique is presented to estimate the Bit error rate from the eye diagram of BER and Q-factor. Accordingly, the proposed method is simulated using Python software. The performance metrics are Q-factor, and bit error rate and the proposed method is compared with the existing Three-Dimensional Single Weight Zero Cross-Correlation (3D-SWZCC) Code, Two Dimensional Spectral/Spatial Cyclic Shift (2D-SSCS), and Permutation Matrix Zero Cross-Correlation (PM-ZCC) Code. Subsequently, these results exhibit that the proposed method produces the best performance than the other methods and it estimates the BER. Consequently, future work is based on the study of 3D codes with m-sequence code in the spectral/time/spatial OCDMA domain. The approach has elegantly integrated the design of an encoder and a decoder, allowing the system to efficiently create code in 3D bipolar mode, respectively.

Constructing cospectral graphs via regular rational orthogonal matrices with level two

Two graphs G and H are cospectral if they share the same spectrum. Constructing cospectral non-isomorphic graphs has been investigated for many years and various constructions are known in the literature, among which, the GM-switching, invented by Godsil and McKay in 1982, is proved to a simple and powerful one. Motivated by this, we address the following problem: “For which graphs G with adjacency matrix A(G), does there exist a rational orthogonal matrix Q (not a permutation matrix) with constant row sum, such that QTA(G)Q is a (0,1)-matrix?” We focus on a special case that 2Q is an integral matrix with one fully indecomposable block. Some partial answers to the above question are given by associating a directed graph ΓG with G. We find that there exists an interesting relationship between the existence of the above Q and the structure of ΓG. Based on the results obtained, we present a new method for the construction of cospectral graphs.

A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

Centralizer matrix algebras and symmetric polynomials of partitions

For a field R of characteristic p≥0 and a matrix c in the full n×n matrix algebra Mn(R) over R, let Sn(c,R) be the centralizer algebra of c in Mn(R). We show that Sn(c,R) is a Frobenius-finite, 1-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension Sn(c,R)⊆Mn(R) is separable and Frobenius. Further, we study the isomorphism problem of invariant matrix algebras. Let σ be a permutation in the symmetric group Σn and cσ the corresponding permutation matrix in Mn(R). We give sufficient and necessary conditions for the invariant algebra Sn(cσ,R) to be semisimple. If R is an algebraically closed field, we establish a combinatoric characterization of when two semisimple invariant R-algebras are isomorphic in terms of the cycle types of permutations.

Construction and the ergodicity properties of dual unitary quantum circuits

We consider one-dimensional quantum circuits of the brickwork type, where the fundamental quantum gate is dual unitary. Such models are solvable: the dynamical correlation functions of the infinite temperature ensemble can be computed exactly. We review various existing constructions for dual unitary gates and we supplement them with new ideas in a number of cases. We discuss connections with various topics in physics and mathematics, including quantum information theory, tensor networks for the AdS/CFT correspondence (holographic error correcting codes), classical combinatorial designs (orthogonal Latin squares), planar algebras, and Yang-Baxter maps. Afterwards we consider the ergodicity properties of a special class of dual unitary models, where the local gate is a permutation matrix. We find an unexpected phenomenon: nonergodic behavior can manifest itself in multisite correlations, even in those cases when the one-site correlation functions are fully chaotic (completely thermalizing). We also discuss the circuits built out of perfect tensors. They appear locally as the most chaotic and most scrambling circuits, nevertheless they can show global signs of nonergodicity: if the perfect tensor is constructed from a linear map over finite fields, then the resulting circuit can show exact quantum revivals at unexpectedly short times. A brief mathematical treatment of the recurrence time in such models is presented in Appendix (written by Roland Bacher and Denis Serre).

End-to-End Learning the Partial Permutation Matrix for Robust 3D Point Cloud Registration

Even though considerable progress has been made in deep learning-based 3D point cloud processing, how to obtain accurate correspondences for robust registration remains a major challenge because existing hard assignment methods cannot deal with outliers naturally. Alternatively, the soft matching-based methods have been proposed to learn the matching probability rather than hard assignment. However, in this paper, we prove that these methods have an inherent ambiguity causing many deceptive correspondences. To address the above challenges, we propose to learn a partial permutation matching matrix, which does not assign corresponding points to outliers, and implements hard assignment to prevent ambiguity. However, this proposal poses two new problems, i.e. existing hard assignment algorithms can only solve a full rank permutation matrix rather than a partial permutation matrix, and this desired matrix is defined in the discrete space, which is non-differentiable. In response, we design a dedicated soft-to-hard (S2H) matching procedure within the registration pipeline consisting of two steps: solving the soft matching matrix (S-step) and projecting this soft matrix to the partial permutation matrix (H-step). Specifically, we augment the profit matrix before the hard assignment to solve an augmented permutation matrix, which is cropped to achieve the final partial permutation matrix. Moreover, to guarantee end-to-end learning, we supervise the learned partial permutation matrix but propagate the gradient to the soft matrix instead. Our S2H matching procedure can be easily integrated with existing registration frameworks, which has been verified in representative frameworks including DCP, RPMNet, and DGR. Extensive experiments have validated our method, which creates a new state-of-the-art performance.

Economic evaluations of audit and feedback interventions: a systematic review

BackgroundThe effectiveness of audit and feedback (A&F) interventions to improve compliance to healthcare guidelines is supported by randomised controlled trials (RCTs) and meta-analyses of RCTs. However, there is currently a...

Alternating sign matrices of finite multiplicative order

We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form P+T, where P is a permutation matrix and T has four non-zero entries, forming a square with entries 1 and −1 in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.

Image encryption based on a novel memristive chaotic system, Grain-128a algorithm and dynamic pixel masking

In this paper, we first propose a memristive chaotic system and implement it by circuit simulation. The chaotic dynamics and various attractors are analysed by using phase portrait, bifurcation diagram, and Lyapunov exponents. In particular, the system has robust chaos in a wide parameter range and the initial value space, which is favourable to the security communication application. Consequently, we further explore its application in image encryption and present a new scheme. Before image processing, the external key is protected by the Grain-128a algorithm and the initial values of the memristive system are updated with the plain image. We not only perform random pixel extraction and masking with the chaotic cipher, but also use them as control parameters for Brownian motion to obtain the permutation matrix. In addition, multiplication on the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) is added to further enhance the cryptography. Finally, the simulation results verify that the proposed image encryption scheme has better performance and higher security, which can effectively resist various attacks.

Deterministic annealing with Potts neurons for multi-robot routing

A deterministic annealing (DA) method is presented for solving the multi-robot routing problem with min–max objective. This is an NP-hard problem belonging to the multi-robot task allocation set of problems where robots are assigned to a group of sequentially ordered tasks such that the cost of the slowest robot is minimized. The problem is first formulated in a matrix form where the optimal solution of the problem is the minimum-cost permutation matrix without any loops. The solution matrix is then found using the DA method is based on mean field theory applied to a Potts spin model which has been proven to yield near-optimal results for NP-hard problems. Our method is bench-marked against simulated annealing and a heuristic search method. The results show that the proposed method is promising for small-medium sized problems in terms of computation time and solution quality compared to the other two methods.