▁n ▁m   −   r   −   1 ▁times ▁in ▁each ▁of ▁its ▁m ▁sets ▁of ▁n ▁parallel ▁( m   −   1 )- dimensional ▁linear ▁sub spaces ▁( or ▁" layers "). ▁Two ▁such ▁latin ▁hyper c ub es ▁of ▁the ▁same ▁order ▁n ▁and ▁class ▁r ▁with ▁the ▁property ▁that , ▁when ▁one ▁is ▁super im posed ▁on ▁the ▁other , ▁every ▁element ▁of ▁the ▁one ▁occurs ▁exactly ▁n m   −   2 r ▁times ▁with ▁every ▁element ▁of ▁the ▁other , ▁are ▁said ▁to ▁be ▁orthogonal . ▁ ▁A ▁set ▁of ▁k   −   m ▁mut ually ▁orthogonal ▁m - dimensional ▁latin ▁hyper c ub es ▁of ▁order ▁n ▁is ▁equivalent ▁to ▁a ▁ 2 -( n , ▁k , ▁n m   −   2 ) ▁orthogonal ▁array , ▁where ▁the ▁indexing ▁columns ▁form ▁an ▁m -( n , ▁m , ▁ 1 ) ▁orthogonal ▁array . ▁ ▁History ▁The ▁concepts ▁of ▁latin ▁squares ▁and ▁mut ually ▁orthogonal ▁latin ▁squares ▁were ▁generalized ▁to ▁latin ▁cub es ▁and ▁hyper c ub es , ▁and ▁orthogonal ▁latin ▁cub es ▁and ▁hyper c ub es ▁by ▁. ▁ ▁generalized ▁these ▁results ▁to ▁strength ▁t . ▁The ▁present ▁notion ▁of ▁orthogonal ▁array ▁as ▁a ▁general ization ▁of ▁these ▁ideas , ▁due ▁to ▁C . ▁R . ▁R ao , ▁appears ▁in ▁. ▁ ▁Other ▁constru ctions ▁ ▁Had am ard ▁matrices ▁ ▁If ▁there ▁exists ▁a ▁Had am ard ▁matrix ▁of ▁order ▁ 4 m , ▁then ▁there