Related papers: Codes and caps from orthogonal Grassmannians
The article presents a survey of results in algebraic vision and multiview geometry. The starting points is the study of projective algebraic varieties critical for scene reconstruction. Initially studied for reconstructing static scenes in…
The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon. His work describes in great detail the role that curvature plays…
Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the…
The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in…
Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special…
We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic…
We propose a new partial decoding algorithm for $h$-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to $1-(\tfrac{k+g}{n})^{\frac{h}{h+1}}$, where $k$ is the dimension, $n$ the…
The Grassmannian space $\Gr$ is the set of all $k-$dimensional subspaces of the vector space~\smash{$\F_q^n$}. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present…
In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space…
Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Pl\"ucker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words.
In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian…
We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $\Gamma(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that $1<k<n-1$. It is…
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann--Pl\"ucker functions. The equivalence is motivated by Cayley's identities expressing principal and almost-principal minors of a…
When data is stored, compressed, or communicated through a media such as cable or air, sources of noise and other parameters such as EMI, crosstalk, and distance can considerably affect the reliability of these data. Error detection and…
We derive a decomposition for the gradient of the innovation loss with respect to the filter gain in a linear time-invariant system, decomposing as a product of an observability Gramian and a term quantifying the ``non-orthogonality"…
We consider linear codes over a field in which the error values are restricted to a subgroup of its unit group. This scenario captures Lee distance codes as well as codes over the Gaussian or Eisenstein integers. Codes correcting restricted…
In [97,99,100], an fl-RDT framework is introduced to characterize \emph{statistical computational gaps} (SCGs). Studying \emph{symmetric binary perceptrons} (SBPs), [100] obtained an \emph{algorithmic} threshold estimate $\alpha_a\approx…
Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, $V_k(\mathbb{R}^N)$ and $Gr(k, \mathbb{R}^N)$ respectively, and benefit from projection onto lower-dimensional Stiefel and Grassmannian manifolds. In this…
A new construction for constant weight codes is presented. The codes are constructed from $k$-dimensional subspaces of the vector space $\F_q^n$. These subspaces form a constant dimension code in the Grassmannian space $\cG_q(n,k)$. Some of…
Codes in the Grassmannian have recently found an application in random network coding. All the codewords in such codes are subspaces of $\F_q^n$ with a given dimension. In this paper, we consider the problem of list decoding of a certain…