Endomorphisms of Cuboidal Hamming Graphs, Latin Hypercuboids of Class $r$, and Mixed MDS Codes

In this paper we investigate the existence of singular endomorphisms of the cuboidal Hamming graph $H(n_1,...,n_d,S)$ over the set $\left[ n_1\right]\times \left[ n_2\right]\times \cdots \times \left[ n_d\right]$, where $\left[ n\right]=\{1,...,n\}$, which is a generalisation of the well-known (cubic) Hamming graph over $\left[ n\right]^{d}$. Two vertices in $H$ are adjacent, if their Hamming distance lies in the set $S$. In this paper $S=\{1,...,r\}$, for some integer $1\leq r\leq d-1$, and we first show that the singular endomorphisms of minimal rank ( which is the size of their image) of $H(n,...,n,S)$ correspond to Latin hypercubes of class $r$ (those were originally defined by Kishen (1950)). Then we generalise those hypercubes to Latin hypercuboids of class $r$. We discuss the existence of these objects, provide constructions and count Latin hypercuboids for small parameters. In the last part, we extend the natural connection between Latin hypercubes of class $r$ and MDS codes to Latin hypercuboids of class $r$ leading to the definition of MDS codes for mixed codes (mixed MDS codes), that is for codes over hypercuboids. Here, we demonstrate the interdependence between graph endomorphisms, Latin hypercuboids and mixed MDS codes.