On optimal ( Z 6 m x Z 6 n , 4 , 1 ) and ( Z 2 m x Z 18 n , 4 , 1 ) difference packings and their related codes

We give a direct construction of a ( Z p x G , { 0 } x G , 4 , 1 ) relative difference family for G is an element of { Z 6 x Z 6 , Z 2 x Z 18 , Z 6 x Z 18 , Z 2 x Z 54 } and every prime p equivalent to 3 ( mod 4 ) with p > 3. These allow us to construct an optimal ( Z 6 m x Z 6 n , 4 , 1 ) difference packing and an optimal ( Z 2 m x Z 18 n , 4 , 1 ) difference packing for every pair of positive integers ( m , n ). The corresponding optimal optical orthogonal signature pattern codes are also obtained.