We determine completely the number of non-isomorphic t−( t+8, t+2,4) designs for t=2, 3 and 4. Altogether there are 13,769,944 2-(10,4,4) designs, 1749 3-(11,5,4) designs and 11 4-(12,6,4) designs. The number of non-isomorphic simple, decomposable and derived 2-(10,4,4) designs are 10,081,743, 7885 and 8978, respectively. The 2- and 3-designs have been enumerated by a new orderly backtrack algorithm with efficient isomorph rejection in a parallel environment. To construct the 4-(12,6,4) designs, we have used a fast extension algorithm on the feasible 3-designs. Various properties such as group orders, orbit sizes and block intersection numbers are tabulated for the derived 3-(11,5,4) and 4-(12,6,4) designs, including information from which the designs can be generated.