Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 \leq u \leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $\mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.
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The insertion-deletion codes was motivated to correct the synchronization errors. In this paper we prove several Singleton type upper bounds on the insdel distances of linear insertion-deletion codes, based on the generalized Hamming weights and the formation of minimum Hamming weight codewords. Our bound are stronger than some previous known bounds. Some or our upper bounds are valid for any fixed ordering of coordinate positions. We apply these upper bounds to some binary cyclic codes with any rearrangement of coordinate positions, binary Reed-Muller codes and one algebraic-geometric code from elliptic curves.
We consider a stationary linear AR($p$) model with unknown mean. The autoregression parameters as well as the distribution function (d.f.) $G$ of innovations are unknown. The observations contain gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is $\gamma n^{-1/2}$ with an unknown $\gamma$, $n$ is the sample size. The assential problem in such situation is to test the normality of innovations. Normality, as is known, ensures the optimality properties of widely used least squares procedures. To construct and study a Pearson chi-square type test for normality we estimate the unknown mean and the autoregression parameters. Then, using the estimates, we find the residuals in the autoregression. Based on them, we construct a kind of empirical distribution function (r.e.d.f.) , which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. Our Pearson's satatistic is the functional from r.e.d.f. Its asymptotic distributions under the hypothesis and the local alternatives are determined by the asymptotic behavior of r.e.d.f. %Therefore, the study of the asymptotic properties of r.e.d.f. is a natural and meaningful task. In the present work, we find and substantiate in details the stochastic expansions of the r.e.d.f. in two situations. In the first one d.f. $ G (x) $ of innovations does not depend on $ n $. We need this result to investigate test statistic under the hypothesis. In the second situation $ G (x) $ depends on $ n $ and has the form of a mixture $ G (x) = A_n (x) = (1-n ^ {- 1/2}) G_0 (x) + n ^ { -1/2} H (x). $ We need this result to study the power of test under the local alternatives.
Lipschitz bandits is a prominent version of multi-armed bandits that studies large, structured action spaces such as the [0,1] interval, where similar actions are guaranteed to have similar rewards. A central theme here is the adaptive discretization of the action space, which gradually ``zooms in'' on the more promising regions thereof. The goal is to take advantage of ``nicer'' problem instances, while retaining near-optimal worst-case performance. While the stochastic version of the problem is well-understood, the general version with adversarial rewards is not. We provide the first algorithm for adaptive discretization in the adversarial version, and derive instance-dependent regret bounds. In particular, we recover the worst-case optimal regret bound for the adversarial version, and the instance-dependent regret bound for the stochastic version. Further, an application of our algorithm to dynamic pricing (where a seller repeatedly adjusts prices for a product) enjoys these regret bounds without any smoothness assumptions.
Decision trees have long been recognized as models of choice in sensitive applications where interpretability is of paramount importance. In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing, and counting sufficient reasons and contrastive explanations. We prove that the set of all sufficient reasons of minimal size for an instance given a decision tree can be exponentially larger than the size of the input (the instance and the decision tree). Therefore, generating the full set of sufficient reasons can be out of reach. In addition, computing a single sufficient reason does not prove enough in general; indeed, two sufficient reasons for the same instance may differ on many features. To deal with this issue and generate synthetic views of the set of all sufficient reasons, we introduce the notions of relevant features and of necessary features that characterize the (possibly negated) features appearing in at least one or in every sufficient reason, and we show that they can be computed in polynomial time. We also introduce the notion of explanatory importance, that indicates how frequent each (possibly negated) feature is in the set of all sufficient reasons. We show how the explanatory importance of a feature and the number of sufficient reasons can be obtained via a model counting operation, which turns out to be practical in many cases. We also explain how to enumerate sufficient reasons of minimal size. We finally show that, unlike sufficient reasons, the set of all contrastive explanations for an instance given a decision tree can be derived, minimized and counted in polynomial time.
We prove the equidistribution of several multistatistics over some classes of permutations avoiding a $3$-length pattern. We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foze" is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and Steingr\'imsson, and makl is one of the statistics defined by Clarke, Steingr\'imsson and Zeng in 1997. These results solve several conjectures posed by Amini in 2018.
In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size, but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of $C^0$ finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditoners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter $\lambda$ in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.
In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element $v,$ one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element $v,$ one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.
In this paper a problem of numerical simulation of hydraulic fractures is considered. An efficient algorithm of solution, based on the universal scheme introduced earlier by the authors for the fractures propagating in elastic solids, is proposed. The algorithm utilizes a FEM based subroutine to compute deformation of the fractured material. Consequently, the computational scheme retains the relative simplicity of its original version and simultaneously enables one to deal with more advanced cases of the fractured material properties and configurations. In particular, the problems of poroelasticity, plasticity and spatially varying properties of the fractured material can be analyzed. The accuracy and efficiency of the proposed algorithm are verified against analytical benchmark solutions. The algorithm capabilities are demonstrated using the example of the hydraulic fracture propagating in complex geological settings.
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.
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