Constructing MRD codes by switching

Abstract

AbstractMaximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on ‐by‐ matrices over a finite field . They are diameter perfect and have the cardinality if . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in if the other parameters (, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.