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Quantum error correction is essential for the development of any scalable quantum computer. In this work we introduce a generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes. We present new $n$-dimensional toric quantum codes, where $n\geq 5$, which are featured by lattice codes and apply the proposed quantum interleaving method to such new $n$-dimensional toric quantum codes. Through the application of this method to these novel $n$-dimensional toric quantum codes we derive new $n$-dimensional quantum burst-error-correcting codes. Consequently, $n$-dimensional toric quantum codes and burst-error-correcting quantum codes are provided offering both a good code rate and a significant coding gain when it comes to toric quantum codes. Another important consequence from the presented $n$-dimensional toric quantum codes is that if the Golomb and Welch conjecture in \cite{perfcodes} regarding the Lee sphere in $n$ dimensions for the respective close packings holds true, then it follows that these $n$-dimensional toric quantum codes are the only possible ones to be obtained from lattice codes. Moreover, such a methodology can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory., Comment: This manuscript is the proposed generalization of the work New Three and Four-Dimensional Toric and Burst-Error-Correcting Quantum Codes

The magnetic textures generated by a perpendicularly applied magnetic field at the ferromagnetic layer of $Co/Al_{2}O_{3}/Nb$ thin film heterostructures are investigated using magneto-optical imaging and micromagnetic simulations. It is observed that the stray field caused by flux avalanches in the superconducting layer prints out a non-trivial in-plane texture in the cobalt layer, which remains stable up to temperatures much above the Nb critical temperature. These textures mimic quite closely the dendritic thermomagnetic flux avalanches that penetrate the Nb layer from its edges. For low cobalt thickness, the filamentary magnetic textures occur in pairs with opposite polarity. The previous in-plane magnetization determines the relative location of the magnetic filaments with respect to the avalanche position. Micromagnetic simulations solving the Landau-Lifshitz-Gilbert equation confirm the interpretation given for the experimental findings., Comment: 13 paginas, 5 figuras, supplementary material

Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers., Comment: 11 pages

Ongoing research and experiments have enabled quantum memory to realize the storage of qubits. On the other hand, interleaving techniques are used to deal with burst of errors. Effective interleaving techniques for combating burst of errors by using classical error-correcting codes have been proposed in several articles found in the literature, however, to the best of our knowledge, little is known regarding interleaving techniques for combating clusters of errors in topological quantum error-correcting codes. Motivated by that, in this work, we present new three and four-dimensional toric quantum codes which are featured by lattice codes and apply a quantum interleaving method to such new three and four-dimensional toric quantum codes. By applying such a method to these new codes we provide new three and four-dimensional quantum burst-error-correcting codes. As a consequence, new three and four-dimensional toric and burst-error-correcting quantum codes are obtained which have better information rates than those three and four-dimensional toric quantum codes from the literature. In addition to these proposed three and four-dimensional quantum burst-error-correcting codes improve such information rates, they can be used for burst-error-correction in errors which are located, quantum data stored and quantum channels with memory., Comment: 16 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2205.13582

A toric quantum error-correcting code construction procedure is presented in this work. A new class of an infinite family of toric quantum codes is provided by constructing a classical cyclic code on the square lattice $\mathbb{Z}_{q}\times \mathbb{Z}_{q}$ for all odd integers $q\geq 5$ and, consequently, new toric quantum codes are constructed on such square lattices regardless of whether $q$ can be represented as a sum of two squares. Furthermore this work supplies for each $q$ the polyomino shapes that tessellate the corresponding square lattices and, consequently, tile the lattice $\mathbb{Z}^{2}$. The channel without memory to be considered for these constructed toric quantum codes is symmetric, since the $\mathbb{Z}^{2}$-lattice is autodual. Moreover, we propose a quantum interleaving technique by using the constructed toric quantum codes which shows that the code rate and the coding gain of the interleaved toric quantum codes are better than the code rate and the coding gain of Kitaev's toric quantum codes for $q=2n+1$, where $n\geq 2$, and of an infinite class of Bombin and Martin-Delgado's toric quantum codes. In addition to the proposed quantum interleaving technique improves such parameters, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory., Comment: Submitted to "Journal of Algebra, Combinatorics, Discrete Structures and Applications"

Well-rounded lattices have been a topic of recent studies with applications in wiretap channels and in cryptography. A lattice of full rank in Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. In this paper, we investigate when lattices coming from polynomials with integer coefficients are well-rounded.

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the Euler-Poincar\'e characteristic of these fibers (also called Milnor's fibers) and indices (topological degree) of appropriated vector fields defined on spheres of radii small or big enough.

Given a $C^2$ semi-algebraic mapping $F: \mathbb{R}^N \rightarrow \mathbb{R}^p,$ we consider its restriction to $W\hookrightarrow \mathbb{R^{N}}$ an embedded closed semi-algebraic manifold of dimension $n-1\geq p\geq 2$ and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection $\frac{F}{\Vert F \Vert}:W\setminus F^{-1}(0)\to S^{p-1}$. Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering $W$ as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of $F$ with the canonical projection $\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}$ and prove that the fibers of $\frac{F}{\Vert F \Vert}$ and $\frac{\pi\circ F}{\Vert \pi\circ F \Vert}$ are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection $\frac{F}{\Vert F \Vert}$ and $W\cap F^{-1}(0).$ Similar formulae are proved for mappings obtained after composition of $F$ with canonical projections.

Due to mushroom development of wireless devices cognitive radio is used to resolve the bandwidth utilization and sacristy problem. The crafty usage of bandwidth in cognitive radio based on error correcting codes is ensured to accomodate un authorized user. This study proposes a transmission model by which a finite sequence of binary cyclic codes constructed by a binary BCH code of length $n=2^{s}-1$, in which all codes have same error correction capability and code rate but sequentially increasing code lengths greater than $n$. Initially all these codes are carrying data of their corresponding primary users. A transmission pattern is planned in the sprit of interweave model deals the transmission parameters; modulation scheme, bandwidth and code rate. Whenever, any of the primary users having mod of transmission, the binary cyclic code, is not using its allocated bandwidth, the user having its data built by binary BCH code enter and exploit the free path as a secondary user. Eventually whenever the primary user with $W$ bandwidth having binary BCH code for its data transmission, change its status as a secondary user, it just requires the bandwidth less than $W$., Comment: 10 pages

This paper describes the process of the conception of a software tool of TELE management. The proposed management tool combines information from two sources: i) the automatic reports produced by the Learning Content Management System (LCMS) Blackboard and ii) the views of students and teachers on the use of the LCMS in the process of teaching and learning. The results show that the architecture of the proposed management tool has the features of a management tool, since its potential to control, to reset and to enhance the use of an LCMS in the process of teaching and learning and teacher training, is shown.

The Portuguese Technological Plan for Education (TPE) was established to modernize schools and to consolidate the role of Information and Communication Technologies (ICT) in order to promote the academic success of students and allow schools to be transformed into technological enriched environments through a significant learning and knowledge building in a participatory, collaborative and sharing logic. With this work we aimed to establish dynamical interactions students-content- teacher in order to overcome a diagnosed students' lack of effort towards studying curriculum chemistry content. Our methodology design is a theoretical and descriptive one, carried out in a secondary school during the 2009/2010 school year, in order to answer the question "How to improve the engagement of K-12 students in chemistry classes?". Students, gathered in small groups were asked to create digital learning resources (DLR) during classes. The teacher assumed the role of the supervisor, coacher and facilitator of every task that had to be taken or chosen by the students. To enhance interaction student-student and student-teacher, a Twitter account and a Ning site were created for the class. Both supported the Social Learning Environment (SLE) that was intended to be created. The data collected led us to satisfactory results in what concerns the goals of the study. The affordances and constraints of SLE as an open architecture that has potential to facilitate collaborative learning are delineated. Future work should focus on mechanisms that allow assessment both of the methodology used and the students' generated content in order to improve students' learning in this environment., Comment: 9 pages, 6 figures; Proceedings of ICERI2011 Conference; 14th-16th November 2011, Madrid, Spain