Group divisible designs with large block sizes

For positive integers n, k with , let , , and . Lee et al. used the inclusion-exclusion principle to show that the triple is a -GDD of type for where (Lee et al. in Des Codes Cryptogr, , 2017). They conjectured that is also a -GDD of type for any integer . In this paper, we use a similar construction and counting principles to show that there is a -GDD of type for any prime power q and any integers k, n with where . Consequently, their conjecture holds. Such a method is also generalized to yield a -GDD of type where and lambda(k) = Pi(k-1)(i=3) (q(n-)q(e+i-1))/(k-2)vertical bar and k + l <= n + 1..