New constant-dimension subspace codes from parallel cosets of optimal Ferrers diagram rank-metric codes and multilevel inserting constructions

Constant-dimension subspace codes (CDCs), a special class of subspace codes, have attracted significant attention due to their applications in network coding. A fundamental research problem of CDCs is to determine the maximum number of codewords under the given parameters. The paper first proposes the construction of parallel cosets of optimal Ferrers diagram rank-metric codes (FDRMCs) by employing the list of CDCs and inverse list of CDCs. Then a new class of CDCs is obtained by combining the parallel cosets of optimal FDRMCs with parallel linkage construction. Next, we present a novel set of identifying vectors and provide a new construction of CDCs via the multilevel constuction. Finally, the coset construction is inserted into the multilevel construction and three classes of large CDCs are provided, one of which is constructed by using new optimal FDRMCs. Our results establish at least 65 new lower bounds for CDCs with larger sizes than the previously best known codes.