Constructing MRD codes by switching

Abstract

Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on (Formula presented.) -by- (Formula presented.) matrices over a finite field (Formula presented.). They are diameter perfect and have the cardinality (Formula presented.) if (Formula presented.). 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 (Formula presented.) if the other parameters ((Formula presented.), the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.