Related papers: Covering Grassmannian Codes: Bounds and Constructi…
The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. K\"{o}tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network…
The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the Grassmannian space can be used for error-correction in random network coding. On the…
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $\cG_q(n,r)$ by subspaces from the Grassmann graph $\cG_q(n,k)$, $k \geq r$, are discussed. The problem is of interest from four points of view: coding…
The finite Grassmannian $\mathcal{G}_{q}(k,n)$ is defined as the set of all $k$-dimensional subspaces of the ambient space $\mathbb{F}_{q}^{n}$. Subsets of the finite Grassmannian are called constant dimension codes and have recently found…
We introduce the concepts of complex Grassmannian codes and designs. Let G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a finite subset of G(m,n) in which few distances occur, while a design is a finite subset of…
Codes in the Grassmannian space have found recently application in network coding. Representation of $k$-dimensional subspaces of $\F_q^n$ has generally an essential role in solving coding problems in the Grassmannian, and in particular in…
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…
Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet…
Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest…
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…
We give a new asymptotic upper bound on the size of a code in the Grassmannian space. The bound is better than the upper bounds known previously in the entire range of distances except very large values.
Let $\Gamma(n,k)$ be the Grassmann graph formed by the $k$-dimensional subspaces of a vector space of dimension $n$ over a field $\mathbb F$ and, for $t\in \mathbb{N}\setminus \{0\}$, let $\Delta_t(n,k)$ be the subgraph of $\Gamma(n,k)$…
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…
Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. Denote by $\Pi[n,k]_q$…
The {\em metric dimension} of a graph $\Gamma$ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph…
Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its…
One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in $\operatorname{PG}(n,q)$, i.e., the set of subspaces of $\mathbb{F}_q^n$,…
For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze…
A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ .…
Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In…