8 research outputs found
LCD Codes from tridiagonal Toeplitz matrice
Double Toeplitz (DT) codes are codes with a generator matrix of the form
with a Toeplitz matrix, that is to say constant on the diagonals
parallel to the main. When is tridiagonal and symmetric we determine its
spectrum explicitly by using Dickson polynomials, and deduce from there
conditions for the code to be LCD. Using a special concatenation process, we
construct optimal or quasi-optimal examples of binary and ternary LCD codes
from DT codes over extension fields.Comment: 16 page
Group rings: Units and their applications in self-dual codes
The initial research presented in this thesis is the structure of the unit group of the group ring Cn x D6 over a field of characteristic 3 in terms of cyclic groups, specifically U(F3t(Cn x D6)). There are numerous applications of group rings, such as topology, geometry and algebraic K-theory, but more recently in coding theory. Following the initial work on establishing the unit group of a group ring, we take a closer look at the use of group rings in algebraic coding theory in order to construct self-dual and extremal self-dual codes.
Using a well established isomorphism between a group ring and a ring of matrices, we construct certain self-dual and formally self-dual codes over a finite commutative Frobenius ring. There is an interesting relationships between the Automorphism group of the code produced and the underlying group in the group ring. Building on the theory, we describe all possible group algebras that can be used to construct the well-known binary extended Golay code.
The double circulant construction is a well-known technique for constructing self-dual codes; combining this with the established isomorphism previously mentioned, we demonstrate a new technique for constructing self-dual codes. New theory states that under certain conditions, these self-dual codes correspond to unitary units in group rings. Currently, using methods discussed, we construct 10 new extremal self-dual codes of length 68.
In the search for new extremal self-dual codes, we establish a new technique which considers a double bordered construction. There are certain conditions where this new technique will produce self-dual codes, which are given in the theoretical results. Applying this new construction, we construct numerous new codes to verify the theoretical results; 1 new extremal self-dual code of length 64, 18 new codes of length 68 and 12 new extremal self-dual codes of length 80.
Using the well established isomorphism and the common four block construction, we consider a new technique in order to construct self-dual codes of length 68. There are certain conditions, stated in the theoretical results, which allow this construction to yield self-dual codes, and some interesting links between the group ring elements and the construction. From this technique, we construct 32 new extremal self-dual codes of length 68.
Lastly, we consider a unique construction as a combination of block circulant matrices and quadratic circulant matrices. Here, we provide theory surrounding this construction and conditions for full effectiveness of the method. Finally, we present the 52 new self-dual codes that result from this method; 1 new self-dual code of length 66 and 51 new self-dual codes of length 68. Note that different weight enumerators are dependant on different values of β. In addition, for codes of length 68, the weight enumerator is also defined in terms of γ, and for codes of length 80, the weight
enumerator is also de ned in terms of α
Hamiltoniens locaux et information quantique en dimensions réduites
Cette thèse exploite les liens profonds entre la physique des systèmes quantiques
locaux, les propriétés non locales de leurs états fondamentaux et le contenu en information
de ces états. Les deux premiers chapitres sont consacrés à l’application des
systèmes quantiques locaux pour les fins d’une tâche informationnelle précise, soit le
calcul quantique. Au terme d’un bref survol de la théorie, nous proposons un patron
pour le calcul quantique universel et évolutif pouvant être réalisé sur une grande
variété de plateformes physiques, et démontrons qu’il est particulièrement résilient
face à un bruit anisotrope. Les quatre derniers chapitres sont pour leur part consacrés
à l’approche informationnelle des systèmes quantiques à corps multiples. Nous
décrivons les principales propriétés des corrélations et de l’intrication dans les états
fondamentaux des systèmes de dimensions réduites les plus courants, en distinguant
systèmes non critiques et systèmes critiques. Nous montrons que ces propriétés sont
fortement modifiées par la présence de frustration géométrique dans les chaînes de
spins. Enfin, nous réalisons une analyse exhaustive des corrélations et de l’intrication
dans les états fondamentaux de deux théories quantiques de champs non triviales.This thesis exploits the deep connections between the physics of local quantum
systems, the nonlocal features in their ground states, and the information content of
these states. The first two chapters are dedicated to the application of local quantum
systems for the purpose of a definite information-theoretical task, namely quantum
computation. After a brief survey of the theory, we propose a scheme for scalable
universal quantum computation that, we argue, could be implemented on a wide
variety of physical platforms, and show that it is particularly resilient to anisotropic
noise. The last four chapters are dedicated to the information-theoretical approach
of many-body quantum systems. We describe the main properties of correlations and
entanglement in the ground states of the most common low-dimensional many-body
systems, distinguishing between noncritical systems and critical ones. We show how
these properties can be dramatically modified by the presence of geometric frustration
in spin chains. Finally, we perform an intensive study of correlations and
entanglement in the ground states of two nontrivial one-dimensional quantum field
theories