Group divisible designs with large block sizes

Abstract

For positive integers n, k with $3\le k\le n$, let $X=\mathbb {F}_{2^n}\setminus \{0,1\}$, ${\mathcal {G}}=\{\{x,x+1\}:x\in X\}$, and ${\mathcal {B}}_k=\left\{ \{x_1,x_2,\ldots ,x_k\}\!\subset \!X:\sum \limits _{i=1}^kx_i=1,\ \sum \limits _{i\in I}x_i\!\ne \!1\ \mathrm{for\ any}\ \emptyset \!\ne \!I\!\subsetneqq \!\{1,2,\ldots ,k\}\right\}$. 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 $k\in \{3,4,5,6,7\}$ 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 $k\ge 8$. 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 $3\le k\le n$ 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 $k+\ell \le n+1$.