scholarly journals Pairwise Balanced Designs From Cyclic PBIB Designs

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
D K Ghosh ◽  
N R Desai ◽  
Shreya Ghosh
Keyword(s):  
Block Design ◽  
Block Designs ◽  
Incomplete Block ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Balanced Incomplete Block

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.

scholarly journals On the relationship between graphs and partially balanced incomplete block designs

1969 ◽  
Vol 1 (3) ◽  
pp. 425-430 ◽  
Author(s):  
W.D. Wallis
Keyword(s):  
Necessary Conditions ◽  
Block Design ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Balanced Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
The Relationship

Certain theorems which are already known show that if a partially balanced incomplete block design with suitable parameters exists then there is a (V, K, Λ)-graph. We prove that the existence of such a graph is in fact equivalent to the existence of a certain partially balanced design. The known necessary conditions for (V, K, Λ)-graphs then follow from well-known necessary conditions for designs.

scholarly journals Resolvable (r, λ)-designs and the Fisher inequality

1979 ◽  
Vol 28 (4) ◽  
pp. 471-478 ◽  
Author(s):  
S. A. Vanstone
Keyword(s):  
Block Design ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Incomplete Block Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Similar Inequality ◽  
Balanced Incomplete Block

AbstractIt is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.

Generalized Hadamard Matrices and Orthogonal Arrays of Strength Two

1964 ◽  
Vol 16 ◽  
pp. 736-740 ◽  
Author(s):  
S. S. Shrikhande
Keyword(s):  
Block Design ◽  
Orthogonal Arrays ◽  
Hadamard Matrices ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Balanced Incomplete Block Designs ◽  
Group Divisible ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.

On Balanced Incomplete Block Designs with Large Number of Elements

1970 ◽  
Vol 22 (1) ◽  
pp. 61-65 ◽  
Author(s):  
Haim Hanani
Keyword(s):  
Block Design ◽  
Necessary Condition ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
Image Position

A balanced incomplete block design (BIBD) B[k, λ; v] is an arrangement of v distinct elements into blocks each containing exactly k distinct elements such that each pair of elements occurs together in exactly λ blocks.The following is a well-known theorem [5, p. 248].THEOREM 1. A necessary condition for the existence of a BIBD B[k, λ,v] is that(1)It is also well known [5] that condition (1) is not sufficient for the existence of B[k, λ; v].There is an old conjecture that for any given k and λ condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.There is an old conjecture that for any given k and X condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.

A Note on Balanced Incomplete Block Designs

1954 ◽  
Vol 6 ◽  
pp. 341-346 ◽  
Author(s):  
D. A. Sprott
Keyword(s):  
Block Design ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block

A balanced incomplete block design is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects. Such designs have been studied for their combinatorial interest, as in (3), and also for their application to statistics, where the objects are usually varieties.

An Embedding Theorem for Balanced Incomplete Block Designs

1954 ◽  
Vol 6 ◽  
pp. 35-41 ◽  
Author(s):  
Marshall Hall ◽  
W. S. Connor
Keyword(s):  
Euclidean Plane ◽  
Block Design ◽  
Finite Projective Plane ◽  
Symmetric Design ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Derived Design ◽  
Balanced Incomplete Block

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

On Covering of Balanced Incomplete Block Designs

1964 ◽  
Vol 16 ◽  
pp. 615-625 ◽  
Author(s):  
Haim Hanani
Keyword(s):  
Block Design ◽  
Necessary Condition ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
Image Position ◽  
Positive Integers

Given a set E of v elements and given positive integers k (k ≤ v) and λ, we understand by balanced incomplete block design (BIBD) B [k, λ, v] a system of blocks (subsets of E) having k elements each such that every pair of elements of E is contained in exactly λ blocks.A necessary condition for the existence of a design B [k, λ, v] is known to be (4)

scholarly journals Construction of balanced incomplete block designs

1977 ◽  
Vol 23 (3) ◽  
pp. 348-353 ◽  
Author(s):  
Elizabeth J. Morgan
Keyword(s):  
Block Design ◽  
Block Designs ◽  
Incomplete Block ◽  
Finite Plane ◽  
Balanced Incomplete Block Design ◽  
Original Design ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block

AbstractGiven a symmetric balanced incomplete block design or a finite plane, we recursively construct balanced incomplete block designs by taking unions of certain blocks and points of the original design to be the blocks of the new design.

scholarly journals Some Series Of Partially Balanced Incomplete Block Designs

1955 ◽  
Vol 7 ◽  
pp. 369-381 ◽  
Author(s):  
D. A. Sprott
Keyword(s):  
Block Design ◽  
Special Kind ◽  
Statistical Technique ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Designs ◽  
Incomplete Block Design ◽  
The Difference ◽  
Balanced Incomplete Block ◽  
Standard Statistical Technique

1. Introduction. The use of incomplete block designs for estimating and judging the significance of the difference of treatment effects is now a standard statistical technique. A special kind of incomplete block design is the Partially Balanced Incomplete Block Design (PBIBD) introduced in (3).

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.

Sign in / Sign up

Export Citation Format

Share Document