On the existence of orthogonal arrays

By an OA ( 3 , 5 , v ) we mean an orthogonal array (OA) of order v, strength t = 3 , index unity and 5 constraints. The existence of such an OA implies the existence of a pair of mutually orthogonal Latin squares (MOLSs) of side v. After Bose, Shrikhande and Parker (1960) [2] disproved the long-standing Euler conjecture in 1960, one has good reason to believe that an OA ( 3 , 5 , 4 n + 2 ) exists for any integer n ⩾ 2 . So far, however, no construction of an OA ( 3 , 5 , 4 n + 2 ) for any value of n has been given. This paper tries to fill this gap in the literature by presenting an OA ( 3 , 5 , 4 n + 2 ) for infinitely many values of n ⩾ 62 .