Column cyclic rank metric codes and linear complementary dual rank metric codes

We introduce and study column cyclic rank metric (CCRM) codes over finite fields. We present natural questions for the existence and characterization of CCRM codes. We completely solve these questions only in the first nontrivial case: The case of the minimum rank distance d = 2 and the number of rows m = 2. We also consider linear complementary dual rank metric (LCDRM) codes both in the setting of CCRM codes and in the general setting of rank metric codes. We also ask the natural questions for the existence and characterization of CCRM codes in the subclass of LCDRM codes. We also completely solve these questions for this subclass of LCDRM codes only in the first nontrivial case that d = 2 and m = 2. Moreover we extend a characterization of linear complementary dual (LCD) codes of Massey to arbitrary rank metric codes in the general setting.