Decomposable super‐simple BIBDs with block size 4 and index 4, 6

Abstract

Decomposable designs are important combinatorial designs, and have applications in constant weight codes, perfect threshold schemes and graph decompositions. Let λ $\lambda $ be an even positive integer. A λ 2 $\frac{\lambda }{2}$‐decomposable super‐simple ( v , k , λ ) $(v,k,\lambda )$‐balanced incomplete block design (BIBD) is a super‐simple ( v , k , λ ) $(v,k,\lambda )$‐BIBD, ( V , ℬ ) $(V,{\rm{ {\mathcal B} }})$, where ℬ = MJX-tex-caligraphicnormalℬ 1 ∪ MJX-tex-caligraphicnormalℬ 2 ${\rm{ {\mathcal B} }}={{\rm{ {\mathcal B} }}}_{1}\cup {{\rm{ {\mathcal B} }}}_{2}$, and each ( V , MJX-tex-caligraphicnormalℬ i ) $(V,{{\rm{ {\mathcal B} }}}_{i})$ is a true( v , k , λ 2 true) $(v,k,\frac{\lambda }{2})$‐BIBDs for i = 1 , 2 $i=1,2$. In this paper, for λ ∈ MathClass-open{ 4 , 6 MathClass-close} $\lambda \in \{4,6\}$, it is proved that the necessary conditions for the existence of a λ 2 $\frac{\lambda }{2}$‐decomposable super‐simple ( v , 4 , λ ) $(v,4,\lambda )$‐BIBD are also sufficient.

The existence of λ $\lambda $‐decomposable super‐simple (4,2λ) $(4,2\lambda )$‐GDDs of type gu ${g}^{u}$ with λ=2,4 $\lambda =2,4$

“…For u = 59 and 62, 2-decomposable super-simple (4, 4)-GDDs of types 15 12 5 1 and 4 7 6 1 exist from Lemma 2.5 [23], and super-simple (4, 2)-GDDs of types 2 4 and 6 4 exist from Lemma 1.2. So 4-decomposable super-simple (4, 8)-GDDs of types 30 24 5 1…”

“…The following results also will be used in the sequel. Lemma 1.3 (Adams et al [5], Chen et al [23,24], and Zhang and Ge [40]). There exists a…”

“…To obtain the main results, we need the following constructions. These constructions and their analogues are used to construct decomposable super-simple designs in [23,24,33,35] In this section, Theorem 1.4 will be proved. For direct constructions of λ-decomposable supersimple λ (4, 2 )-GDDs of type g u with ∈ λ {2, 4}, we will use Z gu as the set of points and take the groups as…”

“…is the required 2-decomposable super-simple (4, 4)-GDD of type 3 6 . For ∈ g {3 , 3 , 3 , 3 , 3 , 6 , 6 } u 7 8 9 10 11 4 5 , the designs are from Lemma 2.4 in [23].…”

“…For the existence of 1‐decomposable super‐simple $$(4,\lambda )$$‐GDDs of type $${g}^{u}$$ with $$\lambda \in \{2,3,4\}$$, we refer the reader to [38, 40]. The necessary conditions for the existence of $$\lambda $$‐decomposable super‐simple $$(v,4,2\lambda )$$‐BIBDs are also sufficient when $$\lambda \in \{1,2,3,4\}$$ (see [5, 23, 24, 33, 35]). For more results on decomposable super‐simple designs, we refer the reader to [5, 38, 39] and the references therein.…”