Ternary designs with the maximum or near maximum number of replication numbers

Abstract T emory designs with replication numbers all or all but one of the values in the range 1\(V-1J/(K-l} to 1\(V-l)/(K-2) are studied. It is known that for K=3 and replication numbers all values in the range 1\{V-l) /(K-l J to 1\(V-l }/(K-2J, designs exist if and only if 1\=2. Here we show the uniqueness of such designs. When K=3 and all values in the range 1\(V-l J/(K-l) to 1\(V-l )/(K-2) but one are used, designs are constructed for 1\=2 and all V except for one case. Nonexistence is shown for the missing case. Ternary designs with even block size and replication numbers all or all but Ofle of the values in the range A(V-l }/(K-l) to 1\(V-l)/(K-2) and temary designs with block size an odd number greater than three, 1\=2, and A(V-l )/(K-l) 1\(V-l )/(K-2):;;2 are also describe::d.