The interplay of different metrics for the construction of constant dimension codes

TL;DR

This paper surveys recent methods for constructing constant dimension codes, introduces a general framework, and presents improved constructions and bounds for specific parameters relevant to network coding and cryptography.

Contribution

It provides a unifying framework for constant dimension code construction, demonstrating potential improvements and offering new bounds and constructions for key parameters.

Findings

Improved constructions for A_q(10,4;5), A_q(11,4;4), A_q(12,6;6), A_q(15,4;4)

Derived general upper bounds for subcodes in these constructions

Surveyed recent developments and potential for further improvements

Abstract

A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2\dim(U\cap W)\ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g.\ random linear network coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases $A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.