Un codi C rep el nom de Z2Z4-additiu si les seves coordenades es poden dividir en dos subconjunts X i Y , de tal manera que el codi punctured de C, obtingut en eliminar les coordenades que no es troben a X –o, respectivament, a Y – és un codi binari lineal – respectivament, un codi quaternari lineal–. La imatge del mapa de Gray de C és un codi binari i, sovint, no lineal, que s’anomena Z2Z4-lineal. Aquesta tesi presenta noves famílies de codis Z2Z4-additius, amb la particularitat que les seves imatges de Gray són codis Z2Z4-lineals que tenen els mateixos paràmetres i propietats que la coneguda família de codis de Reed-Muller binaris i lineals. També, tot considerant la classe de codis perfectes Z2Z4-lineals, els quals se sap que són completament regulars, es fan servir les construccions d’extensió, puncture, shorten i lifting, i s’estudia si els codis obtinguts són uniformement empaquetats o completament regulars. A més de proporcionar fiabilitat en els canals de comunicació, la teoria de codis s’ha aplicat recentment a l’esteganografia, és a dir, a la ciència d’ocultar informació confidencial dins d’altres missatges, aparentment inofensius –l’objecte recobridor–, de manera que terceres parts no puguin detectar l’esmentada informació. Aquest procés s’ha plantejat a la literatura modificant el bit menys significatiu dels símbols de l’objecte recobridor per transmetre el missatge secret (esteganografia binària), o bé modificant els dos bits menys significatius ( 1-esteganografia). Respecte a la 1-esteganografia, s’exposen dos nous mètodes d’embedding basats en codis perfectes Z2Z4-lineals, que assoleixen una taxa d’embedding més alta que amb altres mètodes ja coneguts, per una distorsió donada; mentre que es presenta un altre mètode, basat en el producte de dos o més codis de Hamming q-aris, conforme a l’esteganografia binària. / Un código C recibe el nombre de Z2Z4-aditivo si sus coordenadas se pueden dividir en dos subconjuntos X e Y , tales que el código punctured de C, obtenido a partir de eliminar las coordenadas que no están en X –o, respectivamente, en Y – es un código binario lineal –respectivamente, un código cuaternario lineal–. La imagen del mapa de Gray de C es un código Z2Z4-lineal, que es un código binario y, a menudo, no lineal. En esta tesis se presentan nuevas familias de códigos Z2Z4-aditivos, con la particularidad de que sus imágenes a través del mapa de Gray son códigos Z2Z4-lineales con los mismos parámetros y propiedades que la conocida familia de códigos de Reed-Muller binarios y lineales. Considerando la clase de códigos perfectos Z2Z4-lineales, los cuales se sabe que son completamente regulares, se han utilizado las construcciones de extensión, puncture, shorten y lifting, y estudiado si los códigos obtenidos en cada caso eran uniformemente empaquetados o completamente regulares. Además de proporcionar fiabilidad en los canales de comunicación, la teoria de códigos se ha aplicado recientemente a la esteganografía, es decir, a la ciencia de ocultar información confidencial en otros mensajes, aparentemente inofensivos –el objeto recubridor– de tal manera que dicha información no pueda ser detectada por terceros. Este proceso se ha planteado en la literatura modificando el bit menos significativo de los símbolos del objeto recubridor (esteganografía binaria), o bien modificando los dos bits menos significativos ( 1-esteganografía). Con respecto a la 1-esteganografía, se exponen dos nuevos métodos de embedding basados en códigos perfectos Z2Z4-lineales, que alcanzan una tasa de embedding superior a la de otros métodos anteriores, para una distorsión dada; mientras que se presenta otro método, basado en el producto de dos o más códigos de Hamming q-arios, conforme a la esteganografía binaria. / A code C is said to be Z2Z4-additive if its coordinates can be partitioned into two subsets X and Y , in such a way that the punctured code of C obtained by removing the coordinates outside X –or, respectively, Y – is a binary linear code –respectively, a quaternary linear code–. The Gray map image of C is a binary and often nonlinear code called Z2Z4-linear code. In this dissertation, new families of Z2Z4-additive codes are presented, with the particularity that their Gray map images are Z2Z4-linear codes having the same parameters and properties as the well-known family of binary linear Reed-Muller codes. Considering the class of perfect Z2Z4-linear codes, which are known to be completely regular, we have used the extension, puncture, shorten and lifting constructions, and studied the uniformly packed condition and completely regularity of the obtained codes. Besides providing reliability in communication channels, coding theory has been recently applied to steganography, i.e., the science of hiding sensitive information within an innocuouslooking message –the cover object– in such a way that third parties cannot detect that information. This hiding process has been addressed in the literature either by distorting the least significant bit of symbols in the cover object to transmit the secret message (binary steganography), or by distorting the two least significant bits ( 1-steganography). With respect to 1-steganography, two new embedding methods based on perfect Z2Z4- linear codes are introduced, achieving a higher embedding rate for a given distortion than previous methods; while another method, based on the product of more than two perfect q-ary Hamming codes, is presented conforming to binary steganography.

It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm is able to factor large integers in polynomial time on a quantum computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields, in addition to families of subsystem codes. This dissertation is organized into three parts: Quantum Block Codes. After introducing the theory of quantum block codes, I establish conditions when BCH codes are self-orthogonal (or dual-containing) with respect to Euclidean and Hermitian inner products. In particular, I derive two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum codes, as well as families of quantum codes derived from projective geometries. Subsystem Codes. Subsystem codes form a new class of quantum codes in which the underlying classical codes do not need to be self-orthogonal. I give an introduction to subsystem codes and present several methods for subsystem code constructions. I derive families of subsystem codes from classical BCH and RS codes and establish a family of optimal MDS subsystem codes. I establish propagation rules of subsystem codes and construct tables of upper and lower bounds on subsystem code parameters. Quantum Convolutional Codes. Quantum convolutional codes are particularly well-suited for communication applications. I develop the theory of quantum convolutional codes and give families of quantum convolutional codes based on RS codes. Furthermore, I establish a bound on the code parameters of quantum convolutional codes - the generalized Singleton bound. I develop a general framework for deriving convolutional codes from block codes and use it to derive families of non-catastrophic quantum convolutional codes from BCH codes. The dissertation concludes with a discussion of some open problems.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 120-124).

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 129-144).

Thesis (M.S.)--West Virginia University, 2001. / Title from document title page. Document formatted into pages; contains v, 118, [64] p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 114-118).

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 63-68). Also available in electronic version. Access restricted to campus users.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 94-99).

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A number of techniques have been developed over the last forty years which attempt to solve the problem of recognizing human speech by machine. Although the general problem of unconstrained, speaker independent connected speech recognition is still not solved, some of the methods have demonstrated varying degrees of success on a number of constrained speech recognition tasks. Human speech communication is considered to take place on a number of levels from the acoustic signal through to higher linguistic and semantic levels. At the acoustic level, the recognition process can be divided into time-alignment (the removal of global and local timing differences between the unknown input speech and the stored reference templates) and referencete mplate matching. Little attention seems to have been given to the effective use of acoustic level contextual information to improve the performance of these tasks. In this thesis, a new template matching scheme is developed which addresses this issue and successfully allows the utilization of acoustic level context. The method, based on Bayesian decision theory, is a dynamic time warping approach which incorporates statistical dependencies in matching errors between frames along the entire length of the reference template. In addition, the method includes a speaker compensation technique operating simultaneously. Implementation is carried out using the highly efficient branch and bound algorithm. Speech model storage requirements are quite small as a result of an elegant feature of the recursive matching criterion. Furthermore, a novel method for inferencing the special speech models is introduced. The new method is tested on data drawn from nearly 8000 utterances of the 26 letters of the British English Alphabet spoken by 104 speakers, split almost equally between male and female speakers. Experiments show that the new approach is a powerful acoustic level speech recognizer achieving up to 34% better recognition performance when compared with a conventional method based on the dynamic programming algorithm.