We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the 1-error correcting property of fivefold codes in any...

This paper considers the ubiquitous problem of estimating the state (e.g., position) of an object based on a series of noisy measurements. The standard approach is to formulate this problem as one of measuring the state (or a function of the state) corrupted by additive Gaussian noise. This model assumes both (i) the sensor provides a measurement of the true target (or, alternatively, a separat...

An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a...

In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning performance is optimized. We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS)...

In various penalty/smoothing approaches to solving a linear program, one regularizes the problem by adding to the linear cost function a separable nonlinear function multiplied by a small positive parameter. Popular choices of this nonlinearfunction include the quadraticfunction, the logarithm function, and the x ln(x)-entropy function. Furthermore, the solutions generated by such approaches ma...

In the past, it has generally been assumed that the prohability of undetected error for an (n,k) block code, used solely for error detection on a binary symmetric channel, is upperbounded by 2-‘“-k’. In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquengh...

This paper compares the error bounds of two classes of dominant point detection methods, viz. methods based on reducing a distance metric and methods based on digital straight segments. We highlight using two methods in each class that the error bounds within the same class may vary depending upon the fine details and control parameters of the methods but the essential error bound can be determ...

Radial basis functions are a very powerful tool in multivariate approximation since they are mesh free. In the theory of radial basis functions, the most frequently used error bounds are the one raised by Madych and Nelson in [5] and the one raised by Wu and Schaback in [6], especially the latter. It seems that Wu and Schaback’s error bound is five times more frequently used than that of Madych...

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generaliz...

In an earlier paper the authors studied simplex codes of type α and β over Z4 and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over Z2s . The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes...