Group divisible designs with large block sizes

For positive integers n, k with (3le kle n), let (X=mathbb {F}_{2^n}setminus {0,1}), ({mathcal {G}}={{x,x+1}:xin X}), and ({mathcal {B}}_k=left{ {x_1,x_2,ldots ,x_k}!subset !X:sum limits _{i=1}^kx_i=1, sum limits _{iin I}x_i!ne !1 mathrm{for any} emptyset !ne !I!subsetneqq !{1,2,ldots ,k}right} ). Lee et al. used the inclusion–exclusion principle to show that the triple ((X,{mathcal {G}},{mathcal {B}}_k)) is a ((k,lambda _k))-GDD of type (2^{2^{n-1}-1}) for (kin {3,4,5,6,7}) where (lambda _k=frac{prod _{i=3}^{k-1}(2^n-2^i)}{(k-2)!}) (Lee et al. in Des Codes Cryptogr,  https://doi.org/10.1007/s10623-017-0395-8, 2017). They conjectured that ((X,{mathcal {G}},{mathcal {B}}_k)) is also a ((k,lambda _k))-GDD of type (2^{2^{n-1}-1}) for any integer (kge 8). In this paper, we use a similar construction and counting principles to show that there is a ((k,lambda _k))-GDD of type ((q^2-q)^{(q^{n-1}-1)/(q-1)}) for any prime power q and any integers k, n with (3le kle n) where (lambda _k=frac{prod _{i=3}^{k-1}(q^n-q^i)}{(k-2)!}). Consequently, their conjecture holds. Such a method is also generalized to yield a ((k,lambda _k))-GDD of type ((q^{ell +1}-q^{ell })^{(q^{n-ell }-1)/(q-1)}) where (lambda _k=frac{prod _{i=3}^{k-1}(q^n-q^{ell +i-1})}{(k-2)!}) and (k+ell le n+1).