We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error–locator polynomial is the unique monic minimal polynomial (shortest linear recurrence) of the syndrome sequence and that it can be obtained by Algorithm MR of Norton. When R is a local ring, we show that the syndrome sequence may have ...

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization and local refinemen...

This paper introduces a new local polynomial estimator and develops supporting asymptotic theory for non-parametric regression in the presence of covariate measurement error. We address the measurement error with Cook and Stefanski’s simulation-extrapolation (SIMEX) algorithm. Our method improves on previous local polynomial estimators for this problem by (1) using a bandwidth selection procedu...

Augmented Downhill Simplex Method (ADSM) is introduced here, that is a heuristic combination of Downhill Simplex Method (DSM) with Random Search algorithm. In fact, DSM is an interpretable nonlinear local optimization method. However, it is a local exploitation algorithm; so, it can be trapped in a local minimum. In contrast, random search is a global exploration, but less efficient. Here, rand...

This paper describes a fast and reliable algorithm which computes smooth piecewise polynomial approximations to functions. It adaptively locates the knots by a procedure that has been shown to provide the optimal rate of convergence as the accuracy requirements (and number of knots) increase. Local Hermite interpolation is used which requires that derivatives of the function be known (or estima...

There are many numerous methods for solving large-scale problems in which some of them are very flexible and efficient in both linear and non-linear cases. League championship algorithm is such algorithm which may be used in the mentioned problems. In the current paper, a new play-off approach will be adapted on league championship algorithm for solving large-scale problems. The proposed algori...

We study the average-case hardness of the class NP against deterministic polynomial time algorithms. We prove that there exists some constant μ > 0 such that if there is some language in NP for which no deterministic polynomial time algorithm can decide L correctly on a 1 − (log n)−μ fraction of inputs of length n, then there is a language L′ in NP for which no deterministic polynomial time alg...

Gaussian boson sampling is a candidate model for demonstrating quantum computational advantage in photonic systems. Here, the authors narrow down range of possible architectures such demonstration, by introducing classical algorithm that can simulate optical systems with shallow-depth circuits and local gates polynomial time.