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 in linear block code rate $\rho/\ell$ $\FF_{q^k}$. This natural extension gives rise to rank-metric with $k$ rows, $k\ell$ columns, dimension $\rho$ minimum distance that satisfies Singleton bound. Given specific skeleton code, can be Ferrers diagram by appending zeros on left side so it has length $n-k$. The generalized lift is Grassmannian code. By taking union family codes, $n$, cardinality $\frac{q^n-1}{q^k-1}$, injection anticode upper bound constructed.