New Construction for Constant Dimension Subspace Codes via a Composite Structure

One of the most fundamental topics in subspace coding is to explore the maximal possible value ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$ such that the subspace distance satisfies $\operatorname{d_S}(U,V) = \dim(U+V)-\dim(U\cap V) \geq d$ for any two different $k$-dimensional subspaces $U$ and $V$ in this set. In this paper, we propose a construction for constant dimension subspace codes by inserting a composite structure composing of an MRD code and its sub-codes. Its vast advantage over the previous constructions has been confirmed through extensive examples. At least $49$ new constant dimension subspace codes which exceeds the currently best codes are constructed.