# Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

## TL;DR

This paper investigates asymptotic bounds for constant dimension codes used in network coding, reviews existing bounds, and introduces an improved linkage construction that surpasses previous bounds for certain code parameters.

## Contribution

It provides a refined linkage construction that yields larger constant dimension codes, improving lower bounds beyond the MRD bound for specific parameters.

## Key findings

- Improved lower bounds for constant dimension codes.
- Relations between known upper bounds are clarified.
- Construction exceeds MRD bounds for certain code parameters.

## Abstract

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 with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.