scholarly journals Evaluating Popular Statistical Properties of Incomplete Block Designs: A MATLAB Program Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1281
Author(s):  
Emmanuel Ikechukwu Mba ◽  
Polycarp Emeka Chigbu ◽  
Eugene Chijindu Ukaegbu
Keyword(s):  
Incidence Matrix ◽  
Block Design ◽  
Latin Square ◽  
Statistical Properties ◽  
Binary Representation ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Incomplete Block Design ◽  
Level Of Analysis

Evaluating the statistical properties of a semi-Latin square, and in general, an incomplete block design, is vital in determining the usefulness of the design for experimentation. Improving the procedures for obtaining these statistical properties has been the subject of some research studies and software developments. Many available statistical software that evaluate incomplete block designs do so at the level of analysis of variance but not for the popular A-, D-, E-, and MV-efficiency properties of these designs to determine their adequacy for experimentation. This study presents a program written in the MATLAB environment using MATLAB codes and syntaxes which is capable of computing the A-, D-, E-, and MV-efficiency properties of any n×n/k semi-Latin square and any incomplete block design via their incidence matrices, where N is the number of rows and columns and k is the number of plots. The only input required for the program to compute the four efficiency criteria is the incidence matrix of the incomplete block design. The incidence matrix is the binary representation of an incomplete block design. The program automatically generates the efficiency values of the design once the incidence matrix has been provided, as shown in the examples.

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.

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.

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.

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.

scholarly journals Two-level Cretan matrices constructed using SBIBD

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
N. A. Balonin ◽  
Jennifer Seberry
Keyword(s):  
Incidence Matrix ◽  
Block Design ◽  
Difference Sets ◽  
Russian Literature ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Orthogonal Matrices ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
First Time

AbstractTwo-level Cretan matrices are orthogonal matrices with two elements, x and y. At least one element per row and column is 1 and the other element has modulus ≤ 1. These have been studied in the Russian literature for applications in image processing and compression. Cretan matrices have been found by both mathematical and computational methods but this paper concentrates on mathematical solutions for the first time.We give, for the first time, families of Cretan matrices constructed using the incidence matrix of a symmetric balanced incomplete block design and Hadamard related difference sets.

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.

Applications of design theory for the constructions of MDS matrices for lightweight cryptography

2017 ◽  
Vol 11 (2) ◽  
Author(s):  
Kishan Chand Gupta ◽  
Sumit Kumar Pandey ◽  
Indranil Ghosh Ray
Keyword(s):  
Coding Theory ◽  
Design Theory ◽  
Block Design ◽  
Latin Square ◽  
Latin Squares ◽  
Lightweight Cryptography ◽  
Incidence Matrices ◽  
Incomplete Block Design ◽  
Interesting Connection ◽  
Balanced Incomplete Block

AbstractIn this paper, we observe simple yet subtle interconnections among design theory, coding theory and cryptography. Maximum distance separable (MDS) matrices have applications not only in coding theory but are also of great importance in the design of block ciphers and hash functions. It is nontrivial to find MDS matrices which could be used in lightweight cryptography. In the SAC 2004 paper [In this paper, we explore the connection between the maximum number of ones in bi-regular matrices and the incidence matrices of Balanced Incomplete Block Design (BIBD). In this paper, tools are developed to computeWe observe an interesting connection between Latin squares and bi-regular matrices and study the conditions under which a Latin square becomes a bi-regular matrix and finally construct MDS matrices from Latin squares. Also a lower bound of

scholarly journals Constructing non-orthogonal split-split-plot designs using some resolvable block designs

2020 ◽  
Vol 57 (2) ◽  
pp. 177-194
Author(s):  
Iwona Mejza ◽  
Katarzyna Ambroży-Deręgowska ◽  
Kazuhiro Ozawa ◽  
Stanisław Mejza ◽  
Shinji Kuriki
Keyword(s):  
Block Design ◽  
Square Lattice ◽  
Block Designs ◽  
Incidence Matrices ◽  
Incomplete Block Design ◽  
Split Plot ◽  
Final Design ◽  
Efficiency Factors ◽  
Stratum Efficiency Factors ◽  
General Balance

SummaryWe consider a new method of constructing non-orthogonal (incomplete) split-split-plot designs (SSPDs) for three (A, B, C) factor experiments. The final design is generated by some resolvable incomplete block design (for the factor A) and by square lattice designs for factors B and C using a modified Kronecker product of those designs (incidence matrices). Statistical properties of the constructed designs are investigated under a randomized-derived linear model. This model is strictly connected with a four-step randomization of units (blocks, whole plots, subplots, sub-subplots inside each block). The final SSPD has orthogonal block structure (OBS) and satisfies the general balance (GB) property. The statistical analysis of experiments performed in the SSPD is based on the analysis of variance often used for multistratum experiments. We characterize the SSPD with respect to the stratum efficiency factors for the basic estimable treatment contrasts. The structures of the vectors defining treatment contrasts are also given.

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