The hull of two classical propagation rules and their applications

Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at $http://www.codetables.de$. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table

On Galois self-orthogonal algebraic geometry codes

Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much attention for their rich algebraic structures and wide applications in these years. In this paper, we consider them together and study Galois SO AG codes. A criterion for an AG code being Galois SO is presented. Based on this criterion, we construct several new classes of maximum distance separable (MDS) Galois SO AG codes from projective lines and several new classes of Galois SO AG codes from projective elliptic curves, hyper-elliptic curves and hermitian curves. In addition, we give an embedding method that allows us to obtain more MDS Galois SO codes from known MDS Galois SO AG codes.Comment: 18paper

On MDS Codes With Galois Hulls of Arbitrary Dimensions

The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict conditions can also lead to many new classes of MDS codes with Galois hulls of arbitrary dimensions.Comment: 21 pages,5 table

On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top $p$ eigenvalues, the simplest orthogonalization-free method is to find the best rank-$p$ approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer-Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer-Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures-Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest $k$ eigenvalues for large-scale matrices if the largest $k$ eigenvalues are nearly distributed uniformly

Several classes of Galois self-orthogonal MDS codes

Let $q=p^h$ be an odd prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods of constructing several classes of $e$-Galois self-orthogonal generalized Reed-Solomn codes and extended generalized Reed-Solomn codes with $2e\mid h$. We can determine all possible $e$-Galois self-orthogonal maximum distance separable codes of certain lengths for each even $h$ and odd prime number $p$.Comment: 18 pages, 9 table

New MDS self-dual codes over finite fields \F_{r^2}

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we develop the existing theory and construct six new classes of MDS self-dual codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57\%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes and MDS almost self-dual codes are also constructed.Comment: 16 pages, 3 tabl

Characterization and mass formulas of symplectic self-orthogonal and LCD codes and their application

The object of this paper is to study two very important classes of codes in coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD) codes under the symplectic inner product, involving characterization, constructions, and their application. Using such a characterization, we determine the mass formulas of symplectic SO and LCD codes by considering the action of the symplectic group, and further obtain some asymptotic results. Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD) codes with improved parameters directly compared with Euclidean SO (resp. LCD) codes. Under the symplectic distance, we obtain some additive SO (resp. additive complementary dual) codes with improved parameters directly compared with Hermitian SO (resp. LCD) codes. Further, we also construct many good additive codes outperform the best-known linear codes in Grassl's code table. As an application, we construct a number of record-breaking (entanglement-assisted) quantum error-correcting codes, which improve Grassl's code table

Reliability model of organization management chain of South-to-North Water Diversion Project during construction period

AbstractIn order to analyze the indispensability of the organization management chain of the South-to-North Water Diversion Project (SNWDP), two basic forms (series connection state and mixed state of both series connection and parallel connection) of the organization management chain can be abstracted. The indispensability of each form has been studied and is described in this paper. Through analysis of the reliability of the two basic forms, reliability models of the organization management chain in the series connection state and the mixed state of both series connection and parallel connection have been set up

Lattice arrangement of myosin filaments correlates with fiber type in rat skeletal muscle

The thick (myosin-containing) filaments of vertebrate skeletal muscle are arranged in a hexagonal lattice, interleaved with an array of thin (actin-containing) filaments with which they interact to produce contraction. X-ray diffraction and EM have shown that there are two types of thick filament lattice. In the simple lattice, all filaments have the same orientation about their long axis, while in the superlattice, nearest neighbors have rotations differing by 0 degrees or 60 degrees . Tetrapods (amphibians, reptiles, birds, and mammals) typically have only a superlattice, while the simple lattice is confined to fish. We have performed x-ray diffraction and electron microscopy of the soleus (SOL) and extensor digitorum longus (EDL) muscles of the rat and found that while the EDL has a superlattice as expected, the SOL has a simple lattice. The EDL and SOL of the rat are unusual in being essentially pure fast and slow muscles, respectively. The mixed fiber content of most tetrapod muscles and/or lattice disorder may explain why the simple lattice has not been apparent in these vertebrates before. This is supported by only weak simple lattice diffraction in the x-ray pattern of mouse SOL, which has a greater mix of fiber types than rat SOL. We conclude that the simple lattice might be common in tetrapods. The correlation between fiber type and filament lattice arrangement suggests that the lattice arrangement may contribute to the functional properties of a muscle

Iterative distributed minimum total-MSE approach for secure communications in MIMO interference channels

In this paper, we consider the problem of joint transmit precoding (TPC) matrix and receive filter matrix design subject to both secrecy and per-transmitter power constraints in the MIMO interference channel, where K legitimate transmitter-receiver pairs communicate in the presence of an external eavesdropper. Explicitly, we jointly design the TPC and receive filter matrices based on the minimum total mean-squared error (MT-MSE) criterion under a given and feasible information-theoretic degrees of freedom. More specifically, we formulate this problem by minimizing the total MSEs of the signals communicated between the legitimate transmitter-receiver pairs, whilst ensuring that the MSE of the signals decoded by the eavesdropper remains higher than a certain threshold. We demonstrate that the joint design of the TPC and receive filter matrices subject to both secrecy and transmit power constraints can be accomplished by an efficient iterative distributed algorithm. The convergence of the proposed iterative algorithm is characterized as well. Furthermore, the performance of the proposed algorithm, including both its secrecy rate and MSE, is characterized with the aid of numerical results. We demonstrate that the proposed algorithm outperforms the traditional interference alignment (IA) algorithm in terms of both the achievable secrecy rate and the MSE. As a benefit, secure communications can be guaranteed by the proposed algorithm for the MIMO interference channel even in the presence of a "sophisticated/strong" eavesdropper, whose number of antennas is much higher than that of each legitimate transmitter and receiver