On Hull-Variation Problem of Equivalent Linear Codes

The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. The construction of an entanglement-assisted quantum code from a linear code over ${\bf F}_q$ or ${\bf F}_{q^2}$ depends essentially on the Euclidean hull or the Hermitian hull of this code. Therefore it is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. A general method to construct hull-decreasing or hull-increasing equivalent linear codes is proposed. We prove that for a nonnegative integer $h$ satisfying $0 \leq h \leq n-1$, a linear $[2n, n]_q$ self-dual code is equivalent to a linear $h$-dimension hull code. On the opposite direction we prove that a linear LCD code over ${\bf F}_{2^s}$ satisfying $d\geq 2$ and $d^{\perp} \geq 2$ is equivalent to a linear one-dimension hull code under a weak condition. Several new families of negacyclic LCD codes and BCH LCD codes over ${\bf F}_3$ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new EAQEC codes including MDS and almost MDS entanglement-assisted quantum codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte

A transform approach to polycyclic and serial codes over rings

Producción CientíficaIn this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.Ministerio de Ciencia, Innovación y Universidades / Agencia Estatal de Investigación / 0.13039/501100011033 (grant PGC2018-096446-B-C21

Twisted skew GG-codes

In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are often algebras over a finite field, allows us to detect many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes

Matrix Product Structure of a Permuted Quasi Cyclic Code and Its dual

In my Dissertation I will work mostly with Permuted Quasi Cyclic Codes. They are a generalization of Cyclic Codes, one of the most important families of Linear Codes in Coding Theory. Linear Codes are very useful in error detection and correction. Error Detection and Correction is a technique that first detects the corrupted data sent from some transmitter over unreliable communication channels and then corrects the errors and reconstructs the original data. Unlike linear codes, cyclic codes are used to correct errors where the pattern is not clear and the error occurs in a short segment of the message. The length of Permuted Cyclic Codes usually is a big number, that is why I will try to break them down into cyclic codes of small length. This way we can make the study of these code easier and understand them better. One way of breaking down big codes is to write them down as matrix product of small codes. From any permuted quasi cyclic code, we can define some special cyclic codes. I will try to find a sufficient and necessary conditions so any permuted quasi cyclic code can be written as a matrix product of those codes. Another generalization of cyclic codes is the family of multi cyclic codes. These types of codes are more complicated than the previous one so I will propose to limit myself on finding the structure of ternary multi cyclic codes of length 4. One technique of constructing new linear codes from a given linear code is by finding the so called Euclidean dual of a linear code. In my thesis I will also analyze the Euclidean dual of the families above

Collective Spin-Cavity Ensembles and the Protection of Higher-dimensional Quantum Information

The development of devices leveraging their quantum nature to outperform classical analogues has been an ongoing effort for many decades now. In this thesis we investigate two facets related to this effort, albeit at different potential timescales of utility. First, we discuss our contributions in understanding collective spin ensembles interacting with a resonance cavity. This set up is common in superconducting qubit devices and electron spin-resonance experiments. The traditional model when considering this situation is the Tavis--Cummings model, although many of the methods could be adapted to other mesoscopic systems composed of ensembles of spins and cavities. In particular, we focus on characterizing the shifts in the energies due to dressing states, which are known as Lamb shifts. Before this line of work, most efforts focused on generating difference equations which could be solved iteratively to extract these shifts and dressed states. While this methodology works for systems involving hundreds of spins to thousands of spins, this iterative construction loses utility for larger ensembles due to the time needed to determine the parameters and prevents broad trends from being noted. Through these works we have stated how to determine the moments of the statistical distribution of the Lamb shifts, how to bound the largest of these shifts, and which of the subspaces are most important when finding these shifts. Beyond this, we have found that by including thermal effects we may use the moments of the Lamb shifts to greatly simplify a perturbative expansion to determine values of certain observables in optimal time (in spin ensemble size). These results provide greater insights into this model, provide faster simulation times, and can aide in experimental tests of these devices. Second, we discuss the contributions made in quantum error-correcting codes. The typical formalism used for quantum error-correcting codes is the stabilizer formalism. In our work we have extended this formalism to no longer directly depend on the local-dimension of the quantum computing device. For instance, most devices being currently designed and built run on qubits, which have local-dimension two, while qutrits have local-dimension of three. By removing this local-dimensional dependency we are able to generate many stabilizer codes, including codes with parameters previously unknown--amongst which are local-dimension-invariant forms, with the same distance parameter value, for the Steane code, the entire quantum analog of the classical Hamming family, and the Toric code. While these codes do not outperform the best known codes, this serves as an interesting pedagogical and extended framework and may provide for improved codes upon sufficient consideration, or aide in other work in fields closely related to stabilizers. Meanwhile, this extended framework permits for the importation of quantum error-correcting codes from lower local-dimension values to devices with higher local-dimension values, which at least provides some code options if a quantum computer is developed with easily tuned local-dimension value. These topics should be considered as disjoint, and all variable meanings are reset between the topics

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal