Construction of amicably resolvable Pairwise Balanced Design from row-regular Williamson type matrices

2016 ◽  
Author(s):  
M. K. Singh ◽  
Pinky Pandey
Keyword(s):  
Pairwise Balanced Design ◽  
Balanced Design

scholarly journals Some Pairwise Balanced Designs

10.37236/1491 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
Malcolm Greig
Keyword(s):  
Block Design ◽  
Basis Sets ◽  
Combinatorial Designs ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

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.

An Existence Theory for Incomplete Designs

2016 ◽  
Vol 59 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Peter Dukes ◽  
Esther R. Lamken ◽  
Alan C. H. Ling
Keyword(s):  
Existence Results ◽  
The Other ◽  
Existence Theory ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Incomplete Designs ◽  
Block Sizes

AbstractAn 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.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Manjusri Basu ◽  
Debabrata Kumar Ghosh ◽  
Satya Bagchi
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Fold System

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.

scholarly journals Recursive constructions for equidistant permutation arrays

1977 ◽  
Vol 24 (2) ◽  
pp. 216-223 ◽  
Author(s):  
P. J. Schellenberg ◽  
S. A. Vanstone
Keyword(s):  
Direct Product ◽  
Hamming Distance ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Permutation Arrays ◽  
Product Construction ◽  
Permutation Array ◽  
Recursive Constructions ◽  
Room Squares

AbstractAn equidistant permutation array (EPA) is a ν ×rarray defined on anr-set,R, such that (i) each row is a permutation of the elements ofRand (ii) any two distinct rows agree in λ positions (that is, the Hamming distance is (r−λ)).Such an array is said to have order ν. In this paper we give several recursive constructions for EPA's.The first construction uses a resolvable regular pairwise balanced design of ordervto construct an EPA of order ν. The second construction is a generalization of the direct product construction for Room squares.We also give a construction for intersection permutation arrays, which arrays are a generalization of EPA's.

A MULTIPROCESSOR NETWORK ARRANGING REPLICATED OBJECTS IN PAIRWISE BALANCED COMBINATORIAL DESIGNS

1996 ◽  
Vol 06 (01) ◽  
pp. 85-91 ◽  
Author(s):  
SIMON Y. BERKOVICH ◽  
LIN-CHING CHANG
Keyword(s):  
Distributed Systems ◽  
Simulation Study ◽  
Object Oriented ◽  
Communication Overhead ◽  
Combinatorial Designs ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Direct Support ◽  
Design Property ◽  
New Type

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.

Pairwise Balanced Designs with Block Sizes Three and Four

1991 ◽  
Vol 43 (4) ◽  
pp. 673-704 ◽  
Author(s):  
Charles J. Colbourn ◽  
Alexander Rosa ◽  
Douglas R. Stinson
Keyword(s):  
Necessary Conditions ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

AbstractGiven integers ν, a and b, when does a pairwise balanced design on ν elements with a triples and b quadruples exist? Necessary conditions are developed, and shown to be sufficient for all v ≥ 96. An extensive set of constructions for pairwise balanced designs is used to obtain the result.

scholarly journals On the Wilson monoid of a pairwise balanced design

2020 ◽  
Vol 3 (3) ◽  
pp. 637-665 ◽  
Author(s):  
Stuart Margolis ◽  
John Rhodes ◽  
Pedro V. Silva
Keyword(s):  
Pairwise Balanced Design ◽  
Balanced Design
Sign in / Sign up

Export Citation Format

Share Document