Cardinality-consistent flag codes with longer type vectors

Flag codes generalize constant dimension codes by considering sequences of nested subspaces with prescribed dimensions as codewords. A comprehensive construction, which unites cyclic orbit flag codes, yields two families of flag codes on ${\mathbb{F}}_q^n$ (where $n=sk+h$ with $s\ge 2$ and $0\le h<k$): optimum distance flag codes of the longest possible type vector $(1,2,\dots ,k,n-k,\dots ,n-1)$ and flag codes with longer type vectors $(1,2,\dots ,k+h,2k+h,\dots ,(s-2)k+h,n-k,\dots ,n-1)$. These flag codes achieve the same cardinality ${\sum }_{i=1}^{s-1}{q}^{ik+h}+1$.