Pairwise Balanced Designs From Cyclic PBIB Designs

A pairwise balanced designs was constructed using cyclic partially balanced incomplete block designs with either (λ1 – λ2) = 1 or (λ2 – λ1) = 1. This method of construction of Pairwise balanced designs is further generalized to construct it using cyclic partially balanced incomplete block design when |(λ1 – λ2)| = p. The methods of construction of pairwise balanced designs was supported with examples. A table consisting parameters of Cyclic PBIB designs and its corresponding constructed pairwise balanced design is also included.

An Existence Theory for Incomplete Designs

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size w, and all (other) block sizes equal k, this is denoted IPBD((v;w), k). In addition to congruence restrictions on v and w, there is also a necessary inequality: v > (k − 1)w. This article establishes two main existence results for IPBD((v;w), k): one in which w is fixed and v is large, and the other in the case v > (k −1+∊)w when w is large (depending on ∊). Several possible generalizations of the problemare also discussed.

Pairwise Balanced Design of Order 6n+4 and 2-Fold System of Order 3n+2

In the Steiner triple system, Bose (1939) constructed the 2−(v,3,1) design for v=6n+3 and later on Skolem (1958) constructed the same for v=6n+1. In the literature we found a pairwise balanced design (PBD) for v=6n+5. We also found the 2-fold triple system of the orders 3n and 3n+1. In this paper, we construct a PBD for v=6n+4 and a 2-fold system of the order 3n+2. The second construction completes the 2-fold system for all n∈N.

Some Pairwise Balanced Designs

A pairwise balanced design, $B(K;v)$, is a block design on $v$ points, with block sizes taken from $K$, and with every pair of points occurring in a unique block; for a fixed $K$, $B(K)$ is the set of all $v$ for which a $B(K;v)$ exists. A set, $S$, is a PBD-basis for the set, $T$, if $T=B(S)$. Let $N_{a(m)}=\{n:n\equiv a\bmod m\}$, and $N_{\geq m}=\{n:n\geq m\}$; with $Q$ the corresponding restriction of $N$ to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that $Q_{1(8)}$ is a basis for $N_{1(8)}\setminus E$, where $E$ is a set of 5 definite and 117 possible exceptions. 2. We construct a 78 element basis for $N_{1(8)}$ with, at most, 64 inessential elements. 3. Bennett and Zhu have shown that $Q_{\geq8}$ is a basis for $N_{\geq8}\setminus E'$, where $E'$ is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs.

A MULTIPROCESSOR NETWORK ARRANGING REPLICATED OBJECTS IN PAIRWISE BALANCED COMBINATORIAL DESIGNS

The paper investigates a new type of computer interconnection structure using a combinatorial arrangement with pairwise balanced design property in which the interconnection is organized through replication of corresponding objects. The suggested system provides low access latency and reduces the communication overhead. A simulation study to evaluate the performance of this system is presented. The considered organization provides a direct support of object-oriented constructs in distributed systems.