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A design is said to be super‐simple if the intersection of any two blocks has at most two elements. A design with index tλ $t\lambda $ is said to be λt ${\lambda }^{t}$‐decomposable, if its blocks can be partitioned into nonempty collections Bi ${{\rm{ {\mathcal B} }}}_{i}$, 1≤i≤t $1\le i\le t$, such that each Bi ${{\rm{ {\mathcal B} }}}_{i}$ with the point set forms a design with index λ $\lambda $. In this paper, it is proved that there exists a 23 ${2}^{3}$‐decomposable super‐simple (v,4,6) $(v,4,6)$‐BIBD (balanced incomplete block design) if and only if v≥16 $v\ge 16$ and v≡1(mod3) $v\equiv 1(\,\mathrm{mod}\,\,3)$. [ABSTRACT FROM AUTHOR]

A design is said to be super‐simple if the intersection of any two of its blocks has at most two elements. A design with index tλ $t\lambda $ is said to be λ $\lambda $‐decomposable, if its blocks can be partitioned into nonempty collections ℬi ${{\rm{ {\mathcal B} }}}_{i}$, 1≤i≤t $1\le i\le t$, such that each ℬi ${{\rm{ {\mathcal B} }}}_{i}$ with the point set forms a design with index λ $\lambda $. In this paper, it is proved that for λ∈{2,4} $\lambda \in \{2,4\}$, there exists a λ $\lambda $‐decomposable super‐simple (4,2λ) $(4,2\lambda)$‐GDD of type gu ${g}^{u}$ if and only if u≥4 $u\ge 4$, g(u−2)≥2λ $g(u-2)\ge 2\lambda $ and g(u−1)≡0(mod3) $g(u-1)\equiv \,0\,(\mathrm{mod}\,3)$, except for (g,u,λ)=(3,5,2) $(g,u,\lambda)=(3,5,2)$, and except possibly for (g,u,λ)∈{(2,7,2),(6,5,4)} $(g,u,\lambda)\in \{(2,7,2),(6,5,4)\}$. [ABSTRACT FROM AUTHOR]

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 λ2 $\frac{\lambda }{2}$‐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=ℬ1∪ℬ2 ${\rm{ {\mathcal B} }}={{\rm{ {\mathcal B} }}}_{1}\cup {{\rm{ {\mathcal B} }}}_{2}$, and each (V,ℬi) $(V,{{\rm{ {\mathcal B} }}}_{i})$ is a (v,k,λ2) $(v,k,\frac{\lambda }{2})$‐BIBDs for i=1,2 $i=1,2$. In this paper, for λ∈{4,6} $\lambda \in \{4,6\}$, it is proved that the necessary conditions for the existence of a λ2 $\frac{\lambda }{2}$‐decomposable super‐simple (v,4,λ) $(v,4,\lambda)$‐BIBD are also sufficient. [ABSTRACT FROM AUTHOR]

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index λ is said to be { λ 1 s 1 , λ 2 s 2 , ... , λ r s r } - decomposable , with ∑ i = 1 r s i λ i = λ , if its blocks can be partitioned into nonempty collections B i j , 1 ≤ i ≤ r , 1 ≤ j ≤ s i , such that each B i j with the point set forms a design with index λ i , 1 ≤ λ i < λ. In this paper, it is proved that there exists a 42-decomposable super-simple (v , 4 , 8) -BIBD (balanced incomplete block design) if and only if v ≡ 1 (mod 3) and v ≥ 19. [ABSTRACT FROM AUTHOR]

Abstract: Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple balanced incomplete block design and show that such a design exists if and only if and except possibly when . [Copyright &y& Elsevier]

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2 K into pentagons. A twofold Steiner pentagon system of order v [TSPS( v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS( v) is said to be super-simple if its underlying ( v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS( v); namely, v ≥ 15 and v ≡ 0 or 1 ( mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct. [ABSTRACT FROM AUTHOR]

Abstract: We show the necessary conditions are sufficient for the existence of group divisible designs (or PBIBDs) with block size with three groups of size for any and any two indices with . [Copyright &y& Elsevier]

Abstract: In this paper we study -GDDs of type possessing both the pan-decomposable property introduced by Granville, Moisiadis, Rees, On complementary decompositions of the complete graph, Graphs and Combinatorics 5 (1989) 57–61 and the pan-orientable property introduced by Grüttmüller, Hartmann, Pan-orientable block designs, Australas. J. Combin. 40 (2008) 57–68. We show that the necessary condition for a -GDD satisfying both of these properties, namely (1) , (mod ), and (2) are not both even if is odd are sufficient. When , our designs are super-simple. We also determine the spectrum of -GDDs which are super-simple and possess some of the decomposable/orientable conditions, but are not pan-decomposable or pan-orientable. In particular, we show that the necessary conditions for a super-simple directable -GDD of type are sufficient. [Copyright &y& Elsevier]

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A ( v, k, λ = 2 μ)-BIBD is called T- orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A ( v, k, λ = 2 μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for ( v, 4, λ = 2 μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a ( v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable ( v, 4, λ = 2 μ)-BIBDs with a pan-orientable, pan-decomposable ( w, 4, λ = 2 μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4 w), and the case λ = 12 still has to be investigated. [ABSTRACT FROM AUTHOR]

Abstract: We give some constructions of new infinite families of group divisible designs, GDD, including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for . For there is one missing critical design. If , then the necessary conditions are sufficient for . For each of , and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod12. [Copyright &y& Elsevier]

Abstract: A Steiner pentagon system of order is said to be super-simple if its underlying -BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary condition for the existence of a super-simple ; namely, and or is sufficient, except for , and possibly for . In the process, we also improve an earlier result for the spectrum of super-simple -BIBDs, removing all the possible exceptions. We also give some new examples of Steiner pentagon packing and covering designs (SPPDs and SPCDs). [Copyright &y& Elsevier]

We show the necessary conditions are sufficient for the existence of GDD(n,2,4; λ1, λ2) with two groups and block size four in which every block intersects each group exactly twice (even GDD''s) or in which every block intersects each group in one or three points (odd GDD''s). We give a construction for near 3-resolvable triple systems TS(n,3,6) for every n⩾4, and these are used to provide constructions for several families of GDDs. [Copyright &y& Elsevier]

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that λ(v−1)≡0 mod (k−1), λv(v−1)≡0 mod k(k−1). In this paper we study k=9 with λ=3, 6 and 12. We show that these conditions on v are sufficient, with the possible exceptions of v=177, 345, 385 when λ=3, and v=213 when λ=6. We give constructions of TD3(10,n)s with 13 possible exceptions, namely n∈{5,6,14,20,35,45,55,56,60,78,84,85,102}; we also reduce the number of unknown TDs with block sizes 8 and 9. [Copyright &y& Elsevier]

An enclosing of BIBD(v,3,λ) into BIBD(v+s,3,λ+1) is point-wise minimal if s is the smallest positive integer for which an enclosing is possible. We show that the necessary conditions are sufficient for a minimal point-enclosing of any BIBD(v,3,λ) into BIBD(v+s,3,λ+1) for 1&les;λ&les;6. Some other general results on enclosings are presented. [Copyright &y& Elsevier]

Super-simple designs are used in other constructions, such as coverings, superimposed codes and perfect hash families etc. The existence of super-simple (v , 4 , λ) -BIBDs has been determined for λ = 2 , 3 , 4 , 5 , 6 and 9. When λ = 7 , the necessary conditions of such a design are that v ≡ 1 , 4 (mod 12) and v ≥ 16. In this paper, we show that these necessary conditions are sufficient. [ABSTRACT FROM AUTHOR]