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We prove that the necessary condition, n ? 0 (mod 3), is sufficient for the existence of GDD( n , 2 , 4 ; 3 , 4 ) except possibly for n = 18 . We prove that necessary conditions for the existence of group divisible designs GDD( n , 2 , 4 ; λ 1 , λ 2 ) with equal number of even and odd blocks are sufficient for GDD ( n , 2 , 4 ; 5 n , 7 ( n - 1 ) ) for all n ? 2 , GDD ( 7 s , 2 , 4 ; 5 s , 7 s - 1 ) for all s , GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for t ? 0(mod 2) and GDD ( 5 t + 1 , 2 , 4 ; 2 ( 5 t + 1 ) , 14 t ) for all t . To complete the existence of such GDDs, one needs to construct two more families: GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for all odd t , and GDD ( 35 s + 21 , 2 , 4 ; 5 s + 3 , 7 s + 4 ) for all positive integers s . |
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