More mutually orthogonal latin squares

AbstractWilson's construction for mutually orthogonal Latin squares is generalized. This generalized construction is used to improve known bounds on the function nr (the largest order for which there do not exist r MOLS). In particular we find n7⩽780, n8⩽4738, n9⩽5842, n10⩽7222, n11⩽7478, n12⩽9286, n13⩽9476, n15⩽10632

Quantum multipartite entangled states, classical and quantum error correction

Studying entanglement is essential for our understanding of such diverse areas as high-energy physics, condensed matter physics, and quantum optics. Moreover, entanglement allows us to surpass classical physics and technologies enabling better information processing, computation, and improved metrology. Recently, entanglement also played a prominent role in characterizing and simulating quantum many-body states and in this way deepened our understanding of quantum matter. While bipartite entanglement is well understood, multipartite entanglement is much richer and leads to stronger contradictions with classical physics. Among all possible entangled states, a special class of states has attracted attention for a wide range of tasks. These states are called k-uniform states and are pure multipartite quantum states of n parties and local dimension q with the property that all of their reductions to k parties are maximally mixed. Operationally, in a k-uniform state any subset of at most k parties is maximally entangled with the rest. The k = bn/2c-uniform states are called absolutely maximally entangled because they are maximally entangled along any splitting of the n parties into two groups. These states find applications in several protocols and, in particular, are the building blocks of quantum error correcting codes with a holographic geometry, which has provided valuable insight into the connections between quantum information theory and conformal field theory. Their properties and the applications are however intriguing, as we know little about them: when they exist, how to construct them, how they relate to other multipartite entangled states, such as graph states, or how they connect under local operations and classical communication. With this motivation in mind, in this thesis we first study the properties of k-uniform states and then present systematic methods to construct closed-form expressions of them. The structure of our methods proves to be particularly fruitful in understanding the structure of these quantum states, their graph-state representation and classification under local operations and classical communication. We also construct several examples of absolutely maximally entangled states whose existence was open so far. Finally, we explore a new family of quantum error correcting codes that generalize and improve the link between classical error correcting codes, multipartite entangled states, and the stabilizer formalism. The results of this thesis can have a role in characterizing and studying the following three topics: multipartite entanglement, classical error correcting codes and quantum error correcting codes. The multipartite entangled states can provide a link to find different resources for quantum information processing tasks and quantify entanglement. Constructing two sets of highly entangled multipartite states, it is important to know if they are equivalent under local operations and classical communication. By understanding which states belong to the same class of quantum resource, one may discuss the role they play in some certain quantum information tasks like quantum key distribution, teleportation and constructing optimum quantum error correcting codes. They can also be used to explore the connection between the Antide Sitter/Conformal Field Theory holographic correspondence and quantum error correction, which will then allow us to construct better quantum error correcting codes. At the same time, their roles in the characterization of quantum networks will be essential to design functional networks, robust against losses and local noise.El estudio del entrelazamiento cuántico es esencial para la comprensión de diversas áreas como la óptica cuántica, la materia condensada e incluso la física de altas energías. Además, el entrelazamiento nos permite superar la física y tecnologías clásicas llevando a una mejora en el procesado de la información, la computación y la metrología. Recientemente se ha descubierto que el entrelazamiento desarrolla un papel central en la caracterización y simulación de sistemas cuánticos de muchos cuerpos, de esta manera facilitando nuestra comprensión de la materia cuántica. Mientras que se tiene un buen conocimiento del entrelazamiento en estados puros bipartitos, nuestra comprensión del caso de muchas partes es mucho más limitada, a pesar de que sea un escenario más rico y que presenta un contraste más fuerte con la física clásica. De entre todos los posibles estados entrelazados, una clase especial ha llamado la atención por su amplia gama de aplicaciones. Estos estados se llaman k-uniformes y son los estados multipartitos de n cuerpos con dimensión local q con la propiedad de que todas las reducciones a k cuerpos son máximamente desordenadas. Operacionalmente, en un estado k-uniforme cualquier subconjunto de hasta k cuerpos está máximamente entrelazado con el resto. Los estados k = n/2 -uniformes se llaman estados absolutamente máximamente entrelazados porque son máximamente entrelazados respecto a cualquier partición de los n cuerpos en dos grupos. Estos estados encuentran aplicaciones en varios protocolos y, en particular, forman los elementos de base para la construcción de los códigos de corrección de errores cuánticos con geometría holográfica, los cuales han aportado intuición importante sobre la conexión entre la teoría de la información cuántica y la teoría conforme de campos. Las propiedades y aplicaciones de estos estados son intrigantes porque conocemos poco sobre las mismas: cuándo existen, cómo construirlos, cómo se relacionan con otros estados con entrelazamiento multipartito, cómo los estados grafo, o como se relacionan mediante operaciones locales y comunicación clásica. Con esta motivación en mente, en esta tesis primero estudiamos las propiedades de los estados k-uniformes y luego presentamos métodos sistemáticos para construir expresiones cerradas de los mismos. La naturaleza de nuestros métodos resulta ser muy útil para entender la estructura de estos estados cuánticos, su representación como estados grafo y su clasificación bajo operaciones locales y comunicación clásica. También construimos varios ejemplos de estados absolutamente máximamente entrelazados, cuya existencia era desconocida. Finalmente, exploramos una nueva familia de códigos de corrección de errores cuánticos que generalizan y mejoran la conexión entre los códigos de corrección de errores clásicos, los estados entrelazados multipartitos y el formalismo de estabilizadores. Los resultados de esta tesis pueden desarrollar un papel importante en la caracterización y el estudio de las tres siguientes áreas: entrelazamiento multipartito, códigos de corrección de errores clásicos y códigos de corrección de errores cuánticos. Los estados de entrelazamiento multipartito pueden aportar una conexión para encontrar diferentes recursos para tareas de procesamiento de la información cuántica y cuantificación del entrelazamiento. Al construir dos conjuntos de estados multipartitos altamente entrelazados, es importante saber si son equivalentes entre operaciones locales y comunicación clásica. Entendiendo qué estados pertenecen a la misma clase de recurso cuántico, se puede discutir qué papel desempeñan en ciertas tareas de información cuántica, como la distribución de claves criptográficas cuánticas, la teleportación y la construcción de códigos de corrección de errores cuánticos óptimos. También se pueden usar para explorar la conexión entre la correspondencia holográfica Anti-de Sitter/Conformal Field Theory y códigos de corrección de errores cuánticos, que nos permitiría construir mejores códigos de corrección de errores. A la vez, su papel en la caracterización de redes cuánticas será esencial en el diseño de redes funcionales, robustas ante pérdidas y ruidos locales

Locating and Detecting Arrays for Interaction Faults

The identification of interaction faults in component-based systems has focused on indicating the presence of faults, rather than their location and magnitude. While this is a valuable step in screening a system for interaction faults prior to its release, it provides little information to assist in the correction of such faults. Consequently tests to reveal the location of interaction faults are of interest. The problem of nonadaptive location of interaction faults is formalized under the hypothesis that the system contains (at most) some number d of faults, each involving (at most) some number t of interacting factors. Restrictions on the number and size of the putative faults lead to numerous variants of the basic problem. The relationships between this class of problems and interaction testing using covering arrays to indicate the presence of faults, designed experiments to measure and model faults, and combinatorial group testing to locate faults in a more general testing scenario, are all examined. While each has some definite similarities with the fault location problems for component-based systems, each has some striking differences as well. In this paper, we formulate the combinatorial problems for locating and detecting arrays to undertake interaction fault location. Necessary conditions for existence are established, and using a close connection to covering arrays, asymptotic bounds on the size of minimal locating and detecting arrays are established. A final version of this paper appears in J Comb Optim (2008) 15: 17-48

Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture

An orthomorphism of a finite group $G$ is a bijection $\phi\colon G\to G$ such that $g\mapsto g^{-1}\phi(g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $G$ is abelian, for any $k\geq 2$ dividing $|G|-1$, there exists an orthomorphism of $G$ fixing the identity and permuting the remaining elements as products of disjoint $k$-cycles. We prove this conjecture for all sufficiently large groups.Comment: 34 page

A Conjoint Analysis of Wetland-Based Recreation: A Case Study of Louisiana Waterfowl Hunting.

Conjoint analysis is a recent evolution in mathematical psychology that has been employed extensively in the marketing environment. The technique is concerned with measuring the joint effect of two or more independent variables on the ordering of a dependent variable. Conjoint analysis relates an individual\u27s preferences to a set of prespecified attributes. The objective of conjoint analysis is to decomposed a set of responses to factorially designed stimuli in which the utility of each stimuli attribute can be inferred from the respondents\u27 evaluations of the stimuli. In addition, conjoint analysis and its economic foundations are developed in the context of conventional related market and non-market valuation approaches. Given the multiattribute nature of wetland based activities such as waterfowl hunting, conjoint analysis becomes an attractive approach in estimating the benefits and values derived from wetland based activities. An empirical and economic analysis is presented in which waterfowl hunters\u27 willingness-to-pay for various hunting trip attributes is derived from a rank-ordered logit specification of the indirect utility function. The hunting trip vignettes are developed according to seven different attributes with each attribute varying across three levels using a fractional factorial experiment. The data for the analysis were derived from questionnaires mailed to 7,500 randomly selected individuals who purchased 1990 Louisiana duck stamps. The statistical estimation technique employed in this research was rank-ordered logit via weighted least squares. Weighted least squares was chosen due to the presence of heteroskedasticity and uncertainty regarding the properties of the error term which masks the efficiency of the ordinary least squares regression. A Box-Cox transformation was also employed to test for specification of the functional form. The results indicated that the length of the hunting season, the daily duck bag limit, and the rate of congestion were three significant factors influencing waterfowl hunters\u27 trip rating preferences. In addition, conjoint analysis appears to be a viable technique for analysis of resource based multiattribute activities