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 \(k+\ell \le n+1\).
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The author would like to thank the referees for careful check and many helpful comments.
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Communicated by V. D. Tonchev.
Research supported by NSFC Grants 11431003 and a project funded by the priority academic program development of Jiangsu higher education institutions.
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Ji, L. Group divisible designs with large block sizes. Des. Codes Cryptogr. 86, 2255–2260 (2018). https://doi.org/10.1007/s10623-017-0448-z
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DOI: https://doi.org/10.1007/s10623-017-0448-z