Construction of optimal flag codes by MRD codes

Abstract: Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-Gonz\'{a}lez et al. constructed optimal $(n,\mathcal{A})$-Optimum distance flag codes(ODFC) for $\mathcal {A}\subseteq {1,2,\ldots,k,n-k,\ldots,n-1}$ with $k\in \mathcal A$ and $k\mid n$. In this paper, we introduce a new construction of $(n,\mathcal A)_q$-ODFCs by maximum rank-metric codes, and prove that there is an $(n,\mathcal{A})$-ODFC of size $\frac{q^n-q^{k+r}}{q^k-1}+1$ for any $\mathcal{A}\subseteq{1,2,\ldots,k,n-k,\ldots,n-1}$ with $\mathcal A\cap {k,n-k}\neq\emptyset$, where $r\equiv n\pmod k$ and $0\leq r<k$. Furthermore, when $k>\frac{q^r-1}{q-1}$, this $(n,\mathcal A)_q$-ODFC is optimal. Specially, when $r=0$, Alonso-Gonz\'{a}lez et al.'s result is also obtained. We also gives a characterization of almost optimum distance flag codes, and construct a family of optimal almost optimum flag distance codes.