Related papers: Covering Grassmannian Codes: Bounds and Constructi…
Two families of complementary codes over finite fields $\mathbb{F}_q$ are studied, where $q=r^2$ is square: i) Hermitian complementary dual linear codes, and ii) trace Hermitian complementary dual subfield linear codes. Necessary and…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
We denote by $\mathbb{F}_q$ the field with $q$ elements. A radius-$r$ extended ball with center in a $1$-dimensional vector subspace $V$ of $\mathbb{F}_q^3$ is the set of elements of $\mathbb{F}_q^3$ with Hamming distance to $V$ at most…
Subfield subcodes of algebraic-geometric codes are good candidates for the use in post-quantum cryptosystems, provided their true parameters such as dimension and minimum distance can be determined. In this paper we present new values of…
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Suppose that $\mathscr{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathscr{F}$ is the minimum dimension of a…
This paper considers coding for so-called partially stuck memory cells. Such memory cells can only store partial information as some of their levels cannot be used due to, e.g., wear out. First, we present a new code construction for…
The book is devoted to study so-called irregular subsets of the Grassmannian manifold $G^{n}_{k}(V)$ (this class of sets was introduced by author). In the previous variant of the book we restrict ourself only to the case when $V$ is an…
The construction of Grassmannian codes in some projective space is of highly mathematical nature and requires strong computational power for the resulting searches. In this paper was constructed, using GAP System for Computational Discrete…
We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We…
It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with…
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the…
We derive closed-form expressions for the Bayes optimal decision boundaries in binary classification of high dimensional overlapping Gaussian mixture model (GMM) data, and show how they depend on the eigenstructure of the class covariances,…
We consider (symmetric, non-degenerate) bilinear spaces over a finite field and investigate the properties of their $\ell$-complementary subspaces, i.e., the subspaces that intersect their dual in dimension $\ell$. This concept generalizes…
We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based…
We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$…
There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptographic and communications. This work studies the generalization of cover-free families from sets to…
The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…
We describe the higher weights of the Grassmann codes $G(2,m)$ over finite fields ${\mathbb F}_q$ in terms of properties of Schubert unions, and in each case we determine the weight as the minimum of two explicit polynomial expressions in…
In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords…
Motivated by iterative decoding techniques for the binary erasure channel Hollmann and Tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes. A generic $(r,s)$--erasure correcting set generates for…