The Latin Hypercube  

The latin hypercubes form the components of the hypercube, below the several types of latin hypercubes are defined (definitions might be taken from the lexicon) 

pure  all 1agonals and all nagonals contain all m digits  
regular 
all 1agonals contain all m digits, all nagonals need not contain all digits but do sum up to the magic sum. 

bare  all 1agonals contain all m digits  
zero nagonal  a bare latin hypercube with the main nagonal all '0'  
relevant subject  Hypercube doubling method  
ordered  a pure latin hypercube with the main nagonal in natural ordering  
relevant subject  MultiMagic hypercube  
normalized 
the axes 1agonals contain higher permutations with increasing axis number. in case of equal 1agonal axes, the need to transpose might follow from following 1agonals parrallel to the given axis. 