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Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on m $m$‐by‐n $n$ matrices over a finite field Fq ${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1) ${q}^{m(n-d+1)}$ if m≥n $m\ge n$. 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 m $m$ if the other parameters (n,q $n,\,q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. [ABSTRACT FROM AUTHOR]

The p‐rank of a Steiner triple system (STS) B is the dimension of the linear span of the set of characteristic vectors of blocks of B, over GF(p). We derive a formula for the number of different STSs of order v and given 2‐rank r2, r2