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.

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 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 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 A note on matroids and block designs

1975 ◽  
Vol 20 (1) ◽  
pp. 54-58
Author(s):  
R. A. Main ◽  
D. J. A. Welsh
Keyword(s):  
Block Design ◽  
Regular Structure ◽  
Close Connection ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Incomplete Block Design ◽  
Combinatorial Geometries ◽  
Balanced Incomplete Block

The close connection between certain types of matroids or combinatorial geometries and block designs is well known. The relationships previously discussed have centred on the loose analogy between the blocks of a design and the hyperplanesor flats ot the matroid or geometry. The matroids which arise in this way have had in the main a very tight regular structure. Here we show that theclass of matroids whose bases are the blocks of a design ismuch wider — indeed from Theorem 6 below we obatain a metroid in a canonical way from any balanced incomplete block design in which no pair of blocks differ by exactly one element.

8. Designs and geometry

2016 ◽  
pp. 123-139
Author(s):  
Robin Wilson
Keyword(s):  
Triple System ◽  
Block Design ◽  
Projective Planes ◽  
Latin Squares ◽  
Block Designs ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Orthogonal Latin Squares ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block

Block designs are used when designing experiments in which varieties of a commodity are compared. ‘Designs and geometry’ introduces various types of block design, and then relates them to finite projective planes and orthogonal latin squares. A block design consists of a set of v varieties arranged into b blocks. If each block contains the same number k of varieties, each variety appears in the same number r of blocks, then for every block design we have v × r = b × k. A balanced incomplete-block design is when all pairs of varieties in a design are compared the same number of times. A triple system is when each block has three varieties.

A New Method of Construction of E-optimal Generalized Group Divisible Designs (GGDD)

2008 ◽  
Vol 1 (1) ◽  
pp. 38-42
Author(s):  
Alex Thannippara ◽  
Sreejith V ◽  
S. C. Bagui ◽  
D. K. Ghosh
Keyword(s):  
Block Design ◽  
Group Testing ◽  
New Method ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Group Divisible Designs ◽  
Group Divisible ◽  
Incomplete Block Design ◽  
Generalized Group Divisible Design ◽  
Balanced Incomplete Block

In this article, we develop a new method of construction of E-optimal generalized group divisible designs through group testing designs. Keywords: Balanced Incomplete Block Design (BIBD); Group Divisible (GD); Generalized Group Divisible Design (GDD); E-optimality. © 2009 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v1i1.1697

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.

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