more information on Latin squares and MOLS, see [@Handbook Part III].) It is well known that there exist 3 MOLS of side $n$ for every $n\neq 2,3,6,10$. Interestingly, even though the corresponding set of 3 MOLS is not known to exist (and is known not to for $n=2,3,6$), the existence of a ${\mathrm{GHD}}(6, 18)$ and ${\mathrm{GHD}}(10, 30)$ has been shown by Brickell [@brickell] and Soicher [@Soicher], respectively. A ${\mathrm{GHD}}(1,3)$ exists trivially but it is easily seen that the existence of a ${\mathrm{GHD}}(2,6)$ or ${\mathrm{GHD}}(3,9)$ is impossible.

\[618\] \[Latin\] There exists a ${\mathrm{GHD}}(n,3n)$ if and only if $n =1$ or $n \geq 4$.

GHDs, permutation arrays and codes {#PA-code}
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In [@DezaVanstone], Deza and Vanstone noted $2$-${\mathrm{GHD}}_k(s,v;\lambda)$ are equivalent to a particular type of permutation array.

A [*permutation array*]{} $\mathrm{PA}(s,\lambda;v)$ on a set $S$ of $s$ symbols is an $v \times s$ array such that each row is a permutation of $S$ and any two rows agree in at most $\lambda$ columns. A permutation array in which each pair of rows agree in exactly $\lambda$ symbols is called [*equidistant*]{}. If, in any given column, each symbol appears either $0$ or $k$ times, then a permutation array is said to be [*$k$-uniform*]{}.

