Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1

Troon John Benedict; Onyango Fredrick; Karanjah Anthony

doi:doi:10.11648/j.ajtas.20231201.12

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Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1

Troon John Benedict, Onyango Fredrick, Karanjah Anthony

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 1)

Received: 13 April 2023 Accepted: 27 April 2023 Published: 10 May 2023

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Abstract

Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).

Published in	American Journal of Theoretical and Applied Statistics (Volume 12, Issue 1)
DOI	10.11648/j.ajtas.20231201.12
Page(s)	13-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Symmetric BIBD, Projection Geometry, Perfect Odd Square, Galos Field

References

[1]	Akra, U. P., Akpan, S. S., Ugbe, T. A. and Ntekim, O. E. (2021). Finite Euclidean Geometry Approach for Constructing Balanced Incomplete Block Design (BIBD). Asian Journal of Probability and Statistics. 11 (4): 47-59.
[2]	Alabi, M. A. (2018). Construction of balanced incomplete block design of lattice series I and II. International Journal of Innovative Scientific and Engineering Technologies Research. 2018; 6 (4): 10-22.
[3]	Alam, N. M. (2014). On Some Methods of Construction of Block Designs. I. A. S. R. I, Library Avenue, New Delhi-110012.
[4]	Bose, R. C. (1939), On the construction of balanced incomplete block designs. Annals of Eugenics, Vol. 9, pp. 353–399.
[5]	Bose, R. C., Shrikhande, S. S., and Parker, E. T. (1960). Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics, 12, 189-203.
[6]	Dey, A. (2010). Incomplete Block Designs. World Scientific Publishing Co. Pte. Ltd. Warren Street. U.S.A.
[7]	Greig, M., and Rees, D. H. (2003). Existence of balanced incomplete block designs for many sets of treatments. Discrete Mathematics, 261 (1-3), 299-324.
[8]	Goud T. S. and Bhatra, C. N. Ch. (2016). Construction of Balanced Incomplete Block Designs. International Journal of Mathematics and Statistics Invention. 4 (1) 2321-4767.
[9]	Hinkelmann, K. and Kempthorne, O. (2005). Design and Analysis of Experiments. John Wiley and Sons, Inc., Hoboken, New Jersey.
[10]	Hsiao-Lih, J., Tai-Chang, H. and Babul, M. H. (2007). A study of methods for construction of balanced incomplete block design. Journal of Discrete Mathematical Sciences and Cryptography Vol. 10 (2007), No. 2, pp. 227–243.
[11]	Janardan, M. (2018). Construction of balanced incomplete block design: an application of galois field. Open Science Journal of Statistics and Application. 2013; 5 (3): 32-39.
[12]	Kageyama, S. (1980). On properties of efficiency balanced designs. Communication in Statistics, A 9 (6), 597-616.
[13]	Khare, M. and W. T. Federer (1981). A simple construction procedure for resolvable incomplete block designs for any number of treatments. Biom. J., 23, 121–132.
[14]	Mahanta, J. (2018). Construction of balanced incomplete block design: An application of Galois field. Open Science Journal of Statistics and Application.
[15]	Mandal, B. N. (2015). Linear Integer Programming Approach to Construction of Balanced Incomplete Block Designs. Communications in Statistics - Simulation and Computation, 44: 6, 1405-1411, DOI: 10.1080/03610918.2013.821482.
[16]	Montgomery, D. C. (2019). Design and analysis of experiment. John Wiley and Sons, New York.
[17]	Neil, J. S. (2010). Construction of balanced incomplete block design. Journal of Statistics and Probability. 12 (5); 231–343.
[18]	Wan, Z. X. (2009). Design theory. World Scientific Publishing Company.
[19]	Yasmin, F., Ahmed, R. and Akhtar, M. (2015). Construction of Balanced Incomplete Block Designs Using Cyclic Shifts. Communications in Statistics—Simulation and Computation 44: 525–532. DOI: 10.1080/03610918.2013.784984.
[20]	Yates, F. (1936). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci., 26, 424-445.

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APA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. American Journal of Theoretical and Applied Statistics, 12(1), 13-17. https://doi.org/10.11648/j.ajtas.20231201.12

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. Am. J. Theor. Appl. Stat. 2023, 12(1), 13-17. doi: 10.11648/j.ajtas.20231201.12

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. Am J Theor Appl Stat. 2023;12(1):13-17. doi: 10.11648/j.ajtas.20231201.12

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@article{10.11648/j.ajtas.20231201.12,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {1},
  pages = {13-17},
  doi = {10.11648/j.ajtas.20231201.12},
  url = {https://doi.org/10.11648/j.ajtas.20231201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231201.12},
  abstract = {Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).},
 year = {2023}
}

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TY  - JOUR
T1  - Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/05/10
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231201.12
DO  - 10.11648/j.ajtas.20231201.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 13
EP  - 17
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231201.12
AB  - Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).
VL  - 12
IS  - 1
ER  -

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Author Information

Troon John Benedict

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya
Onyango Fredrick

Department of Mathematics and Actuarial Science, Maseno University, Luanda, Kenya
Karanjah Anthony

Department of Mathematics, Multimedia University, Nairobi, Kenya

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APA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. (2023). Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. American Journal of Theoretical and Applied Statistics, 12(1), 13-17. https://doi.org/10.11648/j.ajtas.20231201.12

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ACS Style

Troon John Benedict; Onyango Fredrick; Karanjah Anthony. Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. Am. J. Theor. Appl. Stat. 2023, 12(1), 13-17. doi: 10.11648/j.ajtas.20231201.12

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AMA Style

Troon John Benedict, Onyango Fredrick, Karanjah Anthony. Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1. Am J Theor Appl Stat. 2023;12(1):13-17. doi: 10.11648/j.ajtas.20231201.12

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@article{10.11648/j.ajtas.20231201.12,
  author = {Troon John Benedict and Onyango Fredrick and Karanjah Anthony},
  title = {Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {1},
  pages = {13-17},
  doi = {10.11648/j.ajtas.20231201.12},
  url = {https://doi.org/10.11648/j.ajtas.20231201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231201.12},
  abstract = {Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).},
 year = {2023}
}

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TY  - JOUR
T1  - Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1
AU  - Troon John Benedict
AU  - Onyango Fredrick
AU  - Karanjah Anthony
Y1  - 2023/05/10
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231201.12
DO  - 10.11648/j.ajtas.20231201.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 13
EP  - 17
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231201.12
AB  - Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).
VL  - 12
IS  - 1
ER  -

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