Treffer: Construction of optimal flag codes by MRD codes.

Flag codes have received a lot of attention due to its application in random network coding. In 2021, Alonso-González et al. constructed optimal (n , A) q -Optimum distance flag codes (ODFC) for A ⊆ { 1 , 2 , ... , k , n - k , ... , n - 1 } with k ∈ A and k ∣ n . In this paper, we introduce a new construction of (n , A) q -ODFCs by maximum rank-metric codes, and prove that there is an (n , A) q -ODFC of size q n - q k + r q k - 1 + 1 for any A ⊆ { 1 , 2 , ... , k , n - k , ... , n - 1 } with A ∩ { k , n - k } ≠ ∅ , where r ≡ n (mod k) and 0 ≤ r < k . Furthermore, when k > q r - 1 q - 1 , this (n , A) q -ODFC is optimal. Specially, when r = 0 , Alonso-González et al.'s result is also obtained. We also give a characterization of almost optimum distance flag codes, and construct a family of optimal almost optimum flag distance codes. [ABSTRACT FROM AUTHOR]

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