Multidimensional Latin bitrades

A subset of the $n$-dimensional $k$-valued hypercube is a unitrade or united bitrade whenever the size of its intersections with the one-dimensional faces of the hypercube takes only the values $0$ and $2$. A unitrade is bipartite or Hamiltonian whenever the corresponding subgraph of the hypercube is bipartite or Hamiltonian. The pair of parts of a bipartite unitrade is an $n$-dimensional Latin bitrade. For the $n$-dimensional ternary hypercube we determine the number of distinct unitrades and obtain an exponential lower bound on the number of inequivalent Latin bitrades. We list all possible $n$-dimensional Latin bitrades of size less than $2^{n+1}$.
A subset of the $n$-dimensional $k$-valued hypercube is a $t$-fold MDS code whenever the size of its intersection with each one-dimensional face of the hypercube is exactly $t$. The symmetric difference of two single MDS codes is a bipartite unitrade. Each component of the corresponding Latin bitrade is a switching component of one of these MDS codes. We study the sizes of the components of MDS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes. Furthermore, each MDS code is shown to embed in a Hamiltonian $2$-fold MDS code.