Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

A Note on Partial Parallel Classes in Steiner Systems

A partial parallel class of blocks of a Steiner system S(t,k,v) is a collection of pairwise disjoint blocks. The purpose of this note is to show that any S(k,k+1,v) Steiner system, with v?k⁴+3k³+k²+1, has a partial parallel class containing at least (v?k+1)/(k+2) blocks.     

Resolvable balanced bipartite designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets, B¹ and B², where ∣Bⁱ∣ = k_i, i = 1 or 2. Two elements are said to be linked in B if and only if they belong to different subsets of B. A balanced bipartite design, BBD(v, k₁, k₂, λ), is an arrangement of v elements into b blocks, each containing k elements such that each element occurs in exactly r blocks and any two distinct elements are linked in exactly λ blocks. A resolvable balanced bipartite design, RBBD(v, k₁, k₂, λ), is a BBD(v, k₁, k₂, λ), the b blocks of which can be divided into r sets which are called complete replications, such that each complete replication contains all the v elements of the design.Necessary conditions for the existence of RBBD(v, 1, k₂, λ) and RBBD(v, n, n, λ) are obtained and it is shown that some of the conditions are also sufficient. In particular, necessary and sufficient conditions for the existence of RBBD(v, 1, k₂, λ), where k₂ is odd or equal to two, and of RBBD(v, n, n, λ), where n is even and 2n ? 1 is a prime power, are given.     

A construction for large sets of disjoint Kirkman triple systems

A Steiner system S(t, k, v) is called i-resolvable, 0 < i < t, if its block set can be partitioned into S(i, k, v). In this paper, a 2-resolvable S(3, 4, v) is used to construct a large set of disjoint Kirkman triple systems of order 3v − 3 (briefly LKTS) and some new orders for LKTS are then obtained. Research supported by Tianyuan Mathematics Foundation of NSFC Grant 10526032 and Natural Science Foundation of Universities of Jiangsu Province Grant 05KJB110111.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Classifications of certain t-designs

The nonextendability and extendability of both a t-(v, k, λ_t) design (with $\text{k ?}\text{v}\text{2}$ and its complementary t-design are discussed in four cases: (A) a t-design is extendable and its complement is not extendable; (B) both a t-design and its complement are extendable; (C) a t-design is not extendable and its complement is extendable; (D) both a t-design and its complement are not extendable. Nontrivial examples for each case are presented. Furthermore, some series of t-designs belonging to cases (A), (B), and (D) are also given.     

A novel use of t-packings to construct d-disjunct matrices

A t-packing is an ordered pair (V,P) where V is a v-set and P is a collection of k-subsets (blocks) of V such that each t-subset of V occurs in at most one block of P. If each t-subset of V occurs in exactly one block of P, then (V,P) is known as a Steiner (t,k,v)-design. In this paper, we explore a novel use of t-packings to construct d-disjunct matrices.     

A symmetric design with parameters 2-(49, 16, 5)

A symmetric 2-design with parameters (v, k, λ) = (49, 16, 5) is constructed. Both this design and its residual, a design with parameters (v, b, r, k, λ) = (33, 48, 16, 11, 5), seem to be new. The derived designs do not have repeated blocks. The group of the design is cyclic of order 15. There is no polarity.     

On the existence of 2-designs S2(2,3,v) without repeated blocks

The necessary condition of 2 (mod 3) is sufficient for the existence of a 2-design S₂(2,3,v) without repeated blocks.     

A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

A Kirkman square with index λ, latinicity μ, block size k, and v points, KS_k(v;μ,λ), is a t×t array (t=λ(v-1)/μ(k-1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v,k,λ)-BIBD. In a series of papers, Lamken established the existence of the following designs: KS₃(v;1,2) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable (v,3,2)-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50-76], KS₃(v;2,4) with two possible exceptions [E.R. Lamken, The existence of KS₃(v;2,4)s, Discrete Math. 186 (1998) 195-216], and doubly near resolvable (v,3,2)-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable (v,3,2)-BIBDs, J. Combin. Designs 2 (1994) 427-440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS₃(v;1,1) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases.     

Extending symmetric designs

If a symmetric 2-design with parameters (v, k, λ) is extendable, then one of the following holds: v = 4λ + 3, k = 2λ + 1; or v = (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.     

Embedding of a pseudo-residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let v, t and λ₁, 0 ? i ? t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ₀, λ₁,…, λ_t; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λ_t blocks, 0 ? i ? t, and (3) for every block A in $\text{U}$, |A| ?K. It is well-known that if K consists of a singleton k, then λ₀,…, λ_{t ? 1} can be determined from υ, t, k and λ_t. Hence, we shall denote a (λ₀,…, λ_t; {k}, υ)t-design by S_λ(t, k, υ), where λ = λ_t. A Möbius plane M is an S₁(3, q + 1, q² + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ₀ = q³ + q ? 1, λ₁ = q² + q, λ₂ = q + 1, λ₃ = 1, K = {q + 1, q, q ? 1}, and υ = q² ? 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition.     

Sets in Abelian groups with distinct sums of pairs

A subset S={s₁,…,sk} of an Abelian group G is called an St-set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S₂-set in G. Let v(k) denote the order of the smallest Abelian group for which s(G)?k. In this article, bounds for s(G) are developed and v(k) is determined for k?15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection.     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Some mutually disjoint steiner systems

A t-design λ; t-d-n is a system of subsets of size d (called blocks) from an n-set S, such that each t-subset from S is contained in precisely λ blocks. A Steiner system S(l, m, n) is a t-design with parameters 1; l-m-n. Two Steiner systems (or t-designs) are disjoint if they share no blocks. A search has been conducted which resulted in discovering 9 mutually disjoint S(5, 8, 24)'s, 24 mutually disjoint S(4, 7, 23)'s, 60 mutually disjoint S(3, 6, 22)'s, and 197 mutually disjoint S(2, 5, 21)'s. Taking unions of several mutually disjoint Steiner systems will then produce t-designs (with varying λ's) on 21, 22, 23, and 24 points.     

On Defining Sets of Full Designs with Block Size Three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d _D of D there exists a minimal defining set d _F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.     

Defining Sets of Full Designs with Block Size Three II

A defining set of a t?(v, k, ??) design is a subcollection of its blocks which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. The unique simple design with parameters ${{t-\left(v, k, \begin{array}{ll}\left(\begin{array}{ll}v-t\\ k-t\end{array}\right)\end{array}\right)}}$ is said to be the full design on v elements. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. The largest known minimal defining set is given. The existence of a continuous section of the spectrum comprising asymptotically 9v ²/50 values is shown. This gives a quadratic length section of continuous spectrum where only a linear section with respect to v was known before.     

Balanced bipartite weighing designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets B¹ and B², where |Bⁱ| = k_i, i = 1 or 2. Two elements of B are said to be linked or n-linked in B if and only if they belong to different subsets or the same subset of B respectively. A balanced bipartite weighing design, (briefly BBWD (υ, k₁, k₂, λ₁)) is an arrangement of υ elements into b blocks, each containing k elements, such that each element occurs in exactly r blocks, any two distinct elements are linked in exactly λ₁ blocks and n-linked in exactly λ₂ blocks.Given fixed k₁ and k₂, there is always a minimal value of λ₁ such that the necessary conditions for the existence of a BBWD are satisfied for same υ. It is proved that in many cases, the necessary conditions are also sufficient. Some general methods for constructing BBWD's as well as a table of all designs with υ ? 13 are obtained.