Abstract: Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called Ferrers diagram rank-metric codes with given ranks (GFRMCs), to generalize the parallel construction and the parallel multilevel construction in [IEEE Trans. Inf. Theory, 66 (2020), 6884–6897]. The lower bounds for GFRMCs are derived from Ferrers diagram rank-metric codes (FDRMCs). Via GFRMCs, the inverse multilevel construction for CDCs is showed. Furthermore, the double multilevel construction, as an effective construction for CDCs, is presented by combining the inverse multilevel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given.

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for (4δ, 2δ, 2δ)q-CDCs is calculated. It is greater than 0.99926 for any prime power q and any δ ≥ 3.

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criterion for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring the subcodes of Gabidulin codes, if each of the rightmost$\delta -1$columns in the Ferrers diagram$\cal F$has at least$n-r$dots, where$r$is taken in a range, then the conditions that an FDRM code in$\cal F$is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.