Decomposable super‐simple RBIBDs with block size 4 and index 6

Abstract

Necessary conditions for the existence of a decomposable super‐simple resolvable (v,4,6)‐BIBD whose two component subdesigns are both resolvable (v,4,3)‐BIBDs are v≡0 (mod 4) and v≥16. In this paper, it is proved that these necessary conditions are sufficient, except possibly for v∈{268,284,292,296}.

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

“…Hence we can apply Construction 3.4 with $$l=4$$ to obtain 3‐decomposable super‐simple $$(4,6)$$‐GDDs of types $$3{6}^{7}1{6}^{1}$$, $$3{6}^{7}3{2}^{1}$$, $$3{6}^{7}1{6}^{1}2{4}^{1}$$, and $$3{6}^{7}1{6}^{1}2{8}^{1}$$. From [35], a 3‐decomposable super‐simple $$(u,4,6)$$‐BIBD exists for each $$u\in \{16,24,28,32,36\}$$; hence we can apply Construction 3.5 with $$b=0$$ to obtain the required 3‐decomposable super‐simple $$(v,4,6)$$‐BIBDs. □…”

“…In [38], it is proved that there exists a 3‐decomposable super‐simple $$(v,4,6)$$‐BIBD which is near‐resolvable whenever $$v\equiv 1\,(\mathrm{mod}\,4)$$ and $$v\ge 17$$. In [35], it is proved that there exists a 3‐decomposable super‐simple $$(v,4,6)$$‐BIBD which is resolvable whenever $$v\equiv 0\,(\mathrm{mod}\,4)$$ and $$v\ge 16$$, with four possible exceptions $$(v=268,284,292,296)$$. The readers may refer [5] for the definitions of a near‐resolvable BIBD and a resolvable BIBD.…”

“…Proof It is clear that one can obtain a 2‐decomposable super‐simple $$(k,\lambda ^{\prime} )$$‐GDD of type $${g}^{k}$$ from a completely reducible TD$${}_{\lambda }(k,g)$$, where $$\lambda ^{\prime} $$ is even and $$\lambda ^{\prime} \le \lambda $$. From lemma 2.2 and remark 2.3 in [35], if $$p$$ is a prime power, $$\lambda \le p$$ and $$k\le p$$, then there exists a super‐simple completely reducible RTD$${}_{\lambda }(k,p)$$. Thus there exists a 2‐decomposable super‐simple $$(4,4)$$‐GDD of type $${g}^{4}$$ for $$g\in \{4,9,13\}$$.…”

“…The necessary conditions $$v\equiv 1$$ or $$4\,(\mathrm{mod}\,12)$$ and $$v\ge 13$$ for the existence of completely reducible super‐simple $$(v,4,2)$$‐BIBDs are also sufficient (see [6]). The necessary conditions $$v\equiv 0$$ or $$1\,(\mathrm{mod}\,4)$$ and $$v\ge 16$$ for the existence of 3‐decomposable super‐simple $$(v,4,6)$$‐BIBDs are also sufficient except possibly for $$v\in \{268,284,292,296\}$$ (see [35, 38]). For more results on decomposable super‐simple designs, we refer the reader to [6, 26, 33, 36, 39, 40, 41, 42] and the references therein.…”

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

“…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…”