▁statement ▁would ▁hold ▁had ▁the ▁first ▁and ▁second ▁columns ▁been ▁used . ▁This ▁is ▁thus ▁an ▁orthogonal ▁array ▁of ▁strength ▁two . ▁ ▁Orth ogonal ▁arrays ▁general ize ▁the ▁idea ▁of ▁mut ually ▁orthogonal ▁latin ▁squares ▁in ▁a ▁tab ular ▁form . ▁These ▁arrays ▁have ▁many ▁connections ▁to ▁other ▁combin atorial ▁designs ▁and ▁have ▁applications ▁in ▁the ▁statistical ▁design ▁of ▁experiments , ▁coding ▁theory , ▁crypt ography ▁and ▁various ▁types ▁of ▁software ▁testing . ▁ ▁Definition ▁ ▁A ▁t -( v , k , λ ) ▁orthogonal ▁array ▁( t ▁ ≤ ▁k ) ▁is ▁a ▁ λ vt ▁× ▁k ▁array ▁whose ▁entries ▁are ▁chosen ▁from ▁a ▁set ▁X ▁with ▁v ▁points ▁such ▁that ▁in ▁every ▁subset ▁of ▁t ▁columns ▁of ▁the ▁array , ▁every ▁t - tuple ▁of ▁points ▁of ▁X ▁appears ▁in ▁exactly ▁ λ ▁rows . ▁ ▁In ▁this ▁formal ▁definition , ▁provision ▁is ▁made ▁for ▁repet ition ▁of ▁the ▁t - tu ples ▁( λ ▁is ▁the ▁number ▁of ▁repe ats ) ▁and ▁the ▁number ▁of ▁rows ▁is ▁determined ▁by ▁the ▁other ▁parameters . ▁ ▁In ▁many ▁applications ▁these ▁parameters ▁are ▁given ▁the ▁following ▁names : ▁ ▁v ▁is ▁the ▁number ▁of ▁levels , ▁ ▁k ▁is ▁the ▁number ▁of ▁factors , ▁▁ λ vt ▁is ▁the ▁number ▁of ▁experimental ▁runs , ▁ ▁t ▁is ▁the ▁strength , ▁and ▁▁ λ ▁is ▁the ▁index . ▁ ▁An ▁orthogonal ▁array ▁is ▁simple ▁if ▁it ▁does ▁not ▁contain ▁any ▁repeated ▁rows . ▁ ▁An ▁orthogonal ▁array ▁is ▁linear ▁if ▁X