Browsing by Subject "minimum distance"
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Item Code constructions and code families for nonbinary quantum stabilizer code(Texas A&M University, 2005-11-01) Ketkar, Avanti UlhasStabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.Item Code design based on metric-spectrum and applications(Texas A&M University, 2005-02-17) Papadimitriou, Panayiotis D.We introduced nested search methods to design (n, k) block codes for arbitrary channels by optimizing an appropriate metric spectrum in each iteration. For a given k, the methods start with a good high rate code, say k/(k + 1), and successively design lower rate codes up to rate k/2^k corresponding to a Hadamard code. Using a full search for small binary codes we found that optimal or near-optimal codes of increasing length can be obtained in a nested manner by utilizing Hadamard matrix columns. The codes can be linear if the Hadamard matrix is linear and non-linear otherwise. The design methodology was extended to the generic complex codes by utilizing columns of newly derived or existing unitary codes. The inherent nested nature of the codes make them ideal for progressive transmission. Extensive comparisons to metric bounds and to previously designed codes show the optimality or near-optimality of the new codes, designed for the fading and the additive white Gaussian noise channel (AWGN). It was also shown that linear codes can be optimal or at least meeting the metric bounds; one example is the systematic pilot-based code of rate k/(k + 1) which was proved to meet the lower bound on the maximum cross-correlation. Further, the method was generalized such that good codes for arbitrary channels can be designed given the corresponding metric or the pairwise error probability. In synchronous multiple-access schemes it is common to use unitary block codes to transmit the multiple users? information, especially in the downlink. In this work we suggest the use of newly designed non-unitary block codes, resulting in increased throughput efficiency, while the performance is shown not to be substantially sacrificed. The non-unitary codes are again developed through suitable nested searches. In addition, new multiple-access codes are introduced that optimize certain criteria, such as the sum-rate capacity. Finally, the introduction of the asymptotically optimum convolutional codes for a given constraint length, reduces dramatically the search size for good convolutional codes of a certain asymptotic performance, and the consequences to coded code-division multiple access (CDMA) system design are highlighted.Item Deconvolution in Random Effects Models via Normal Mixtures(2010-10-12) Litton, Nathaniel A.This dissertation describes a minimum distance method for density estimation when the variable of interest is not directly observed. It is assumed that the underlying target density can be well approximated by a mixture of normals. The method compares a density estimate of observable data with a density of the observable data induced from assuming the target density can be written as a mixture of normals. The goal is to choose the parameters in the normal mixture that minimize the distance between the density estimate of the observable data and the induced density from the model. The method is applied to the deconvolution problem to estimate the density of $X_{i}$ when the variable $% Y_{i}=X_{i}+Z_{i}$, $i=1,\ldots ,n$, is observed, and the density of $Z_{i}$ is known. Additionally, it is applied to a location random effects model to estimate the density of $Z_{ij}$ when the observable quantities are $p$ data sets of size $n$ given by $X_{ij}=\alpha _{i}+\gamma Z_{ij},~i=1,\ldots ,p,~j=1,\ldots ,n$, where the densities of $\alpha_{i} $ and $Z_{ij}$ are both unknown. The performance of the minimum distance approach in the measurement error model is compared with the deconvoluting kernel density estimator of Stefanski and Carroll (1990). In the location random effects model, the minimum distance estimator is compared with the explicit characteristic function inversion method from Hall and Yao (2003). In both models, the methods are compared using simulated and real data sets. In the simulations, performance is evaluated using an integrated squared error criterion. Results indicate that the minimum distance methodology is comparable to the deconvoluting kernel density estimator and outperforms the explicit characteristic function inversion method.