On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

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.     

EXISTENCE OF OPTIMAL STRONG PARTIALLY BALANCED DESIGNS

A strong partially balanced design SPBD（v, b, k; λ,0） whose b is the maximum number of blocks in all SPBD（v, b, k; λ, 0）, as an optimal strong partially balanced design, briefly OSPBD（v, k, λ） is studied. In investigation of authentication codes it has been found that the strong partially balanced design can be used to construct authentication codes. This note investigates the existence of optimal strong partially balanced design OSPBD（v, k, 1） for k = 3 and 4, and shows that there exists an OSPBD（v, k, 1） for any v ≥ k.     

Some bounds for partially balanced block designs

Bounds on eigenvalues of theC-matrix for a partially balanced block (PBB) design are given together with some bounds on the number of blocks. Furthermore, a certain equiblock-sized PBB design is characterized. These results contain, as special cases, the known results for variance-balanced block designs and so on.     

A note on partially balanced ternary designs

Two new methods of constructing a series of partially balanced ternary designs are presented. One from a BIB design and a PBIB design, and the second from a PBIB design alone, obtained by method of differences in both the cases.     

Existence of resolvable optimal strong partially balanced designs

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0) whose b is the maximum number of blocks in all SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei [Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205-224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v,3,1). We show that there exists an ROSPBD(v,3,1) for any v?3 except v=6,12.     

High-rate LDPC codes from partially balanced incomplete block designs

Journal of Algebraic Combinatorics - This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since...     

Bounds for the number of common symbols in balanced and certain partially balanced designs

We secure inequalities on submatrices of N′N, where N is the incidence matrix of certain designs. These inequalities are then compared with other results previously known.     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

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.     

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.     

On balanced claw designs of complete multi-partite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of a balanced claw design BCD(m, n, c, λ) of a complete m-partite graph λK_m(n, n,…,n) is λ(m - 1)n ≡ 0 (mod 2c) and (m - 1)n ? c.     

On Steiner 3-wise balanced designs of order 17

We determine all S(3, κ, 17)'s which either; (i) have a block of size at least 6; or (ii) have an automorphism group order divisible by 17, 5, or 3; or (iii) contain a semi-biplane; or (iv) come from an S(3, κ, 16) which is not an S(3, 4, 16). There is an S(3, κ, 17) with |G| = n if and only if n ϵ {2^a3^b: 0 ≤ a ≤ 7,0 ≤ b ≤ 1} ∪ {18, 60, 144, 288, 320, 1920, 5760, 16320}. We also search the S(3, κ, 17)'s listed in the appendix for subdesigns S(2, κ, 17) and generate 22 nonisomorphic S(3, κ, 18)'s by adding a new point to such a subdesign. © 1997 John Wiley & Sons, Inc.     

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.     

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.