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The problem of finding code distance has been long studied for the generic ensembles of linear codes and led to several algorithms that substantially reduce exponential complexity of this task. However, no asymptotic complexity bounds are…
Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical…
We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We…
In this article, we solve some of the geometry problems of the N\'aboj 2023 competition with the help of a computer, using examples that the software tool GeoGebra Discovery can calculate. In each case, the calculation requires symbolic…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
We present ErasureHead, a new approach for distributed gradient descent (GD) that mitigates system delays by employing approximate gradient coding. Gradient coded distributed GD uses redundancy to exactly recover the gradient at each…
The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that…
To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional…
This work considers a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a fraction f of the total number of choices in an unsorted data base. An infinite family of…
We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym$(n)$. In particular, we compute orbits of ordered pairs on Sym$(n)$ acted upon by conjugation…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
A new type of algorithms is presented that combine the advantages of quantum and classical ones. Those combined advantages along with aspects of Geometric Algebra that open possibilities unavailable to both of these computations are…
In [4] Camps-Moreno et al. treated (relative) generalized Hamming weights of codes from extended norm-trace curves and they gave examples of resulting good asymmetric quantum error-correcting codes employing information on the relative…
Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\sigma,\delta]$ a skew polynomial ring. Using skew polynomials $f\in R$, we construct division algebras and a generalization of maximum rank distance codes…
The compression of geometric structures is a relatively new field of data compression. Since about 1995, several articles have dealt with the coding of meshes, using for most of them the following approach: the vertices of the mesh are…
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools.…
The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an…
Symmetry breaking is a crucial technique in modern combinatorial solving, but it is difficult to be sure it is implemented correctly. The most successful approach to deal with bugs is to make solvers certifying, so that they output not just…
Here we present some revised arguments to a randomized algorithm proposed by Sudan to find the polynomials of bounded degree agreeing on a dense fraction of a set of points in $\mathbb{F}^{2}$ for some field $\mathbb{F}$.
Bosonic codes with rotational symmetry are currently one of the best performing quantum error correcting codes. Little is known about error propagation and code distance for these rotation codes in contrast with qubit codes and Bosonic…