A new construction of anticode-optimal Grassmannian codes

In this paper, we consider the well-known unital embedding from $\FF_{q^k}$ into $M_k(\FF_q)$ seen as a map of vector spaces over $\FF_q$ and apply this map in a linear block code of rate $\rho/\ell$ over $\FF_{q^k}$. This natural extension gives rise to a rank-metric code with $k$ rows, $k\ell$ columns, dimension $\rho$ and minimum distance $k$ that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length $n-k$. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length $n$, cardinality $\frac{q^n-1}{q^k-1}$, minimum injection distance $k$ and dimension $k$ that satisfies the anticode upper bound can be constructed.