A 2-$(v,k,\lambda)$ design is super-simple if any two blocks intersect in at most two elements of $V$ and it is simple if it has no repeated blocks. When $k=3$, a super-simple design is just a simple design. In [5] it is shown that for each admissible value of $v$, there exists a simple 2-$(v,3,1)$-DD whose smallest defining sets have at least half of the blocks. Moreover, for all $v\equiv1\ ({\rm mod}\ 3)$ there exist a super-simple 2-$(v,4,1)$-DD and a super-simple 2-$(v,4, 2)$-DD whose smallest defining sets have at least half of the blocks [1], [3]. Also, the existence of super-simple 2-$(v, 5, 1)$-DDs and their smallest defining sets for all $v\equiv1,5\ ({\rm mod}\ 10)$ is shown in [2].

Let $K$ be a set of positive integers smaller than or equal to $v$. A $(K,\lambda)$ directed group divisible design (DGDD) of type ${g_1}^{u_1}{g_2}^{u_2}\dots {g_{\ell}}^{u_{\ell}}$ with $ \sum_{i=1}^{\ell}{g_i}{u_i}=v$, is a triple $(V,\mathcal{G},\mathcal{B})$, where $V$ is a $v$-set and $\mathcal{G}$ is a collection of subsets (groups), each of cardinality in $\{g_1,\dots, g_{\ell}\}$, which partition $V$ into $u_1$ groups of size $g_1,\dots,u_{\ell}$ groups of size $g_{\ell}$ and $\mathcal{B}$ is a collection of blocks of $V$ such that if $B\in\mathcal{B}$, then $|B|\in K$. As before, each block is ordered and every pair of distinct elements of $V$ appears in precisely $\lambda$ blocks or one group but not in both. If $\lambda=1$, $(K,1)$-DGDD is denoted by $K$-DGDD.

Recently, the construction of state-of-the-art codes based on the trade analysis of super-simple directed designs has attracted some attention. This application of directed designs and their corresponding trade designs motivated us to investigate a necessary and sufficient condition to have super-simple $(4,1)$-DGDD. We prove that the necessary condition to have these designs is also sufficient. The defining set related to most of the proposed designs contains at least half of the blocks. Moreover, we provide a number of new super-simple $(4,2)$-DGDDs whose defining sets have at least half of the blocks.

References

[1] Amirzade, F. and Soltankhah, N.: Smallest defining sets of super-simple 2-$(v,4,1)$ directed designs. Utilitas Math. 96, 331--344 (2015)

[2] Amirzade, F. and Soltankhah, N.: On super-simple 2-$(v, 5, 1)$ directed designs and their smallest defining sets. Australas. J. Combin. 54, 85--106 (2012)

[3] Boostan, M., Golalizadeh, S. and Soltankhahh, N.: Super-simple 2-$(v, 4, 2)$ directed designs and lower bound for the minimum size of their defining set. Discrete Appl. Math. 54, 14--23 (2016)

[4] Colbourn, C. J. and Dinitz, J. H.: The CRC Handbook of Combinatorial Designs. Boca Raton FL, USA: CRC Press (1996)

[5] Grannell, M. J., Griggs, T. S. and Quinn, K. A. S.: Smallest defining sets of directed triple systems. Discrete Math. 309, 4810--4818 (2009)

[6] Mahmoodian, E. S. and Soltankhah, N.: On defining sets of directed designs. Australas. J. Combin. 19, 179--190 (1999)

[7] Soltankhah, N.: On directed trades Australas. J. Combin. 11, 59--66 (1995)