Coset Constructions of Constant Dimension Codes by Cosets of Optimal Ferrers Diagrams Rank Metric Codes

Abstract

Constant dimension codes (CDCs) have received a lot of attention due to their application in random network coding. One main problem with CDCs is to improve the lower bound of $A_{q}(n,d,k)$ for given parameters $n,d$ and k, where $A_{q}(n,d,k)$ denotes the maximum size of all $(n,M,d,k)_{q}$ CDCs. The paper aims to construct CDCs by combining the coset and linkage construction. Precisely, we first combine the coset and linkage construction in different ways and then turn our attention to the coset construction. To enlarge the size of CDCs constructed from the coset construction, we are devoted to constructing lists of CDCs with fixed distance having size as large as possible by the cosets of optimal Ferrers diagram rank metric codes and the parallelisms in ${\mathcal {G}}_{q}(n, k)$ . As applications, numerous CDCs with larger size than the previously best known codes are obtained, including $A_{q}(18, 6,9), A_{q}(14, 6, 7), ~A_{q}(12, 4, 6), A_{q}(10, 4, 5),A_{q}(14, 4, 7),$ $A_{q}(16, 4, 8)$ and $A_{q}(n, 4,4)$ for $13\leq n\leq 16$ .