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11.12.2017 | Ausgabe 10/2018

# Group divisible designs with large block sizes

Zeitschrift:
Designs, Codes and Cryptography > Ausgabe 10/2018
Autor:
Lijun Ji
Wichtige Hinweise
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.

## Abstract

For positive integers nk 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 kn 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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