We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes...

The hull AK90],,AK92, p. 43] of a linear code is deened to be its intersection with its dual. We give here the number of distinct q-ary linear codes which have a hull of given dimension. We will prove that, asymptotically, the proportion of q-ary codes whose hull has dimension l is a positive constant that only depends on l and q, and consequently that the average dimension of the hull is asymp...

Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to...

A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that subspace distance satisfies $d_S(U,W):=2k-2\dim(U\cap W)\ge d$ all pairs different codewords $U$, $W$. Constant have applications e.g.\ random linear network coding, cryptography, and distributed storage. Bounds a...

In classical coding theory, different types of extendability results of codes are known. A classical example is the result stating that every (4, q − 1, 3)-code over an alphabet of order q is extendable to a (4, q, 3)-code. A constant dimension subspace code is a set of (k− 1)-spaces in the projective space PG(n− 1, q), which pairwise intersect in subspaces of dimension upper bounded by a speci...

Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Plücker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words.

Let V ∼= Fq be a v-dimensional vector space over the finite field Fq with q elements. By [ V k ] we denote the set of all k-dimensional subspaces in V . Its size is given by the q-binomial coefficient [ v k ]q = ∏k−1 i=0 qv−qi qk−qi for 0 ≤ k ≤ v and 0 otherwise. The set of all subspaces of V forms a metric space associated with the so-called subspace distance dS(U,W ) = dim(U + W ) − dim(U ∩ W...

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irredu...

Since the seminal paper of Kötter and Kschischang [20] there is a still growing interest in subspace codes, which are sets of subspaces of the Fq-vector space F n q . If all subspaces, which play the role of the codewords, have the same dimension, say k, then one speaks of constant dimension codes. The most commonly used distance measures for subspace codes, motivated by an information-theoreti...

Based on ideas of Kötter and Kschischang [6] we use constant dimension subspaces as codewords in a network. We show a connection to the theory of q-analogues of a combinatorial designs, which has been studied in [1] as a purely combinatorial object. For the construction of network codes we successfully modified methods (construction with prescribed automorphisms) originally developed for the q-...