Four pairwise balanced designs

We construct pairwise balanced designs on 49, 57, 93, and 129 points of index unity, with block sizes 5, 9, 13, and 29. This completes the determination of the unique minimal finite basis for the PBD-closed set which consists of the integers congruent to 1 modulo 4. The design on 129 points has been used several times by a number of different authors but no correct version has previously appeared in print.     

Finite embedding theorems for partial pairwise balanced designs

The pair (P, p) is a (partial) (n, b)-PBD if (P, p) isa (partial) pairwise balanced design with the property that |P| = n and each block in p has exactly b elements. The following theorems are proved.Theorem. If (P, p) is an (n, b)-PBD and n > b ? 4, then (P, p) has an isomorphic disjoint mate. (Theorem 2.3)Theorem. Suppose k and b are positive integers and b ? 5. There is a constant C(k, b) such that if (P, p) is an (n, b)-PBD and n > C(k, b), then there exist k mutually disjoint isomorphic mates of (P, p). (Theorem 2.2)Theorem. Suppose k and b are positive integers, k ? 2 and b ? 5. If (P, p₁, (P, p₂),…, (P, p_k) is a collection of partial (|P|, b)-PBD's, there exist k (n, b)-PBD's (X, x₁), (X, x₂), …, (X, x_k) such that (P, p₁) is embedded in (X, x₁) and for i ≠ j, p₁ ∩ p₁ = x₁ ∩ x₁. Additionally the existence of certain collections valuable in embedding is explored. (Theorem 4.10)     

On the structure of regular pairwise balanced designs

In this paper we obtain determinantal conditions necessary for the existence of (r,λ)-designs. The work is based on a paper of Connor [2]. In [3] Deza establishes an inequality which must be satisfied by the column vectors of an equidistant code; or, equivalently, the block sizes in an (r,λ)-design. We obtain a generalization of this inequality.     

Uniformly resolvable pairwise balanced designs with blocksizes two and three

A uniformly resolvable pairwise balanced design is a pairwise balanced design whose blocks can be resolved into parallel classes in such a way that all blocks in a given parallel class have the same size. We are concerned here with designs in which each block has size two or three, and we prove that the obvious necessary conditions on the existence of such designs are also sufficient, with two exceptions, corresponding to the non-existence of Nearly Kirkman Triple Systems of orders 6 and 12.     

A note on the structure of pairwise balanced designs

We show that if a pairwise balanced design with λ = 1 has ν objects and block sizes between 2 and ν − 2, and it is not a symmetric balanced design, then each of its largest blocks has a nonempty intersection with at least ν other blocks     

Golay complementary array pairs

Constructions and nonexistence conditions for multi-dimensional Golay complementary array pairs are reviewed. A construction for a d-dimensional Golay array pair from a (d + 1)-dimensional Golay array pair is given. This is used to explain and expand previously known constructive and nonexistence results in the binary case.      

Embeddability and construction of affine α-resolvable pairwise balanced designs

An affine α-resolvable PBD of index λ is a triple (V, B, R), where V is a set (of points), B is a collection of subsets of V (blocks), and R is a partition of B (resolution), satisfying the following conditions: (i) any two points occur together in λ blocks, (ii) any point occurs in α blocks of each resolution class, and (iii) |B| = |V| + |R| − 1. Those designs embeddable in symmetric designs are described and two infinite series of embeddable designs are constructed. The analog of the Bruck–Ryser–Chowla theorem for affine α-resolvable PBDs is obtained. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:111–129, 1998     

Some imbedding theorems for block designs balanced with respect to pairs

We prove imbedding theorems for block designs balanced with respect to pairs, and with the aid of these theorems we establish the existence of (v, k, )-resolvable BIB block designs with parameters v, k, such that =k–1 [and also such that =(k–l)/2 if k is odd], k ¦(p–1) for each prime divisor p of the number v/k; we also establish an imbedding theorem for Kirkman triple systems.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 173–184, July, 1974.     

On equivalence of negaperiodic Golay pairs

Associated pairs as defined by Ito (J Algebra 234:651–663, 2000) are pairs of binary sequence of length 2t satisfying certain autocorrelation properties that may be used to construct Hadamard matrices of order 4t. More recently, Balonin and Dokovi? (Inf Control Syst 5:2–17, 2015) use the term negaperiodic Golay pairs. We define extended negaperiodic Golay pairs and prove a one-to-one correspondence with central relative (4t, 2, 4t, 2t)-difference sets in dicyclic groups of order 8t. We present a new approach for computing negaperiodic Golay pairs up to equivalence, and determine conditions where equivalent pairs correspond to equivalent Hadamard matrices. We complete an enumeration of negaperiodic Golay pairs of length 2t for \(1 \le t \le 10\), and sort them into equivalence classes. Some structural properties of negaperiodic Golay pairs are derived.     

An existence theory for pairwise balanced designs,III: Proof of the existence conjectures

Given positive integers k and λ, balanced incomplete block designs on v points with block size k and index λ exist for all sufficiently large integers v satisfying the congruences λ(v ? 1) ≡ 0 (mod k ? 1) and λv(v ? 1) ≡ 0 (mod k(k ? 1)). Analogous results hold for pairwise balanced designs where the block sizes come from a given set K of positive integers. We also observe that the number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).     

Another class of balanced graph designs: balanced circuit designs

A block considered as a set of elements together with its adjacency matrix A is called a C-block if A is the adjacency matrix of a circuit. A balanced circuit design with parameters v, b, r, k, λ (briefly, BCD(v, k, λ)) is an arrangement of v elements into bC-blocks such that each C-block contains k elements, each element occurs in exactly rC-blocks and any two distinct elements are linked in exactly λ C-blocks.We investigate conditions for the existence of BCD and show, in particular, that if the block-size k ? 6 and the trivial necessary conditions are satisfied, then the corresponding BCD exists.     

Quaternary Golay sequence pairs I: even length

The origin of all 4-phase Golay sequences and Golay sequence pairs of even length at most 26 is explained. The principal techniques are the three-stage construction of Fiedler, Jedwab and Parker involving multi-dimensional Golay arrays, and a ??sum?Cdifference?? construction that modifies a result due to Eliahou, Kervaire and Saffari. The existence of 4-phase seed pairs of lengths 3, 5, 11, and 13 is assumed; their origin is considered in (Gibson and Jedwab, Des Codes Cryptogr, 2010).     

Partial geometric designs and two-class partially balanced designs

It is shown that a partial geometric design with parameters (r, k, t, c) satisfying certain conditions is equivalent to a two-class partially balanced incomplete block design. This generalizes a result concerning partial geometric designs and balanced incomplete block designs.     

Quaternary Golay sequence pairs II: odd length

A 4-phase Golay sequence pair of length s ?? 5 (mod 8) is constructed from a Barker sequence of the same length whose even-indexed elements are prescribed. This explains the origin of the 4-phase Golay seed pairs of length 5 and 13. The construction cannot produce new 4-phase Golay sequence pairs, because there are no Barker sequences of odd length greater than 13. A partial converse to the construction is given, under the assumption of additional structure on the 4-phase Golay sequence pair of length s ?? 5 (mod 8).     

Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

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.     

Balanced nested designs and balanced arrays

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4.