Construction Procedure for Non-trivial T-designs

John Chibayi; David Alila; Fredrick Onyango

doi:doi:10.11648/j.ajtas.20170601.17

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Construction Procedure for Non-trivial T-designs

John Chibayi, David Alila, Fredrick Onyango

Received: 20 July 2016 Accepted: 8 August 2016 Published: 22 February 2017

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Abstract

A t-design is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3…) occurs. The problem of finding all parameters (t, v, k, λt) for which t-(v, k, λt) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner t-designs are known for t ≥ 6 when v > k. The objective of this study therefore to develop new methods of constructing t-designs with t ≥ 3 and λ ≥1. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs.

DOI	10.11648/j.ajtas.20170601.17
Published in	American Journal of Theoretical and Applied Statistics (Volume 6, Issue 1, January 2017)
Page(s)	52-60
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

Block Designs, Steiner Systems, T-designs

References

[1]	Blanchard, J. L (1995a). A construction for Steiner 3-designs, Journal of combinatorial Theory A, 71, 60-67.
[2]	Blanchard, J. L. (1995b). An extension theorem for Steiner systems, Discrete Mathematics 141, no. 1-3, 23-35.
[3]	Blanchard, J. L (1995c). A construction for orthogonal arrays with strength t ≥ 3, Discrete math 137, no. 1-3, 35-44.
[4]	Bogart, K. P. (1990). Introductory Combinatorics second edition, Harcourt Brace Bose, Jovanovich, Inc. Orlando, Florida.
[5]	Cameron, P. J., Maimani, H. R., Omidi, G. R., and Tayfeh-Rezaie, B. (2006). 3-designs PGL (2, q), Discrete Mathematics, 306, vol.23, 3063-3073.
[6]	Colbourn, C. J. et al. (2002). Orthogonal arrays of strength three from regular 3-wise balanced designs.
[7]	Dinitz, J. H., & Stinson, D. R. (1992). Contemporary Design Theory: A collection ofsurveys, Wiley-Interscience.
[8]	Hartman, A. (1994). The fundamental construction for 3-designs. Discrete math 124, no.1-3, 107-131.
[9]	Magliveras, S. S., Kramer, E. S., & Stinson, D. R. (1993). Some new Large sets of t-designs. Australasian journal of Combinatorics 7, pp 189-193.
[10]	Mohácsy, H., and Ray-Chaudhuri, D. K. (2001). A construction for infinite families of Steiner 3-designs, Journal of Combinatorial Theory A, 94, 127-141.
[11]	Mohácsy, H., and Ray-Chaudhuri, D. K. (2002). Candelabra Systems and designs, Journalof Statistical planning and Inference, 106, 419-448.
[12]	Mohácsy, H., and Ray-Chaudhuri, D. K. (2003). A construction for group divisible t-designs with strength t ≥ 2 and index unity, Journal of Statistical planning and Inference, 109, 167-177.
[13]	Onyango, O. F. (2010). Construction of t-(v, k, λt) designs, Journal of mathematical science, vol. 21 no. 4 pp 521-526.
[14]	Qiu-rong Wu. (1991). A note on extending t-designs, Australasia. Journal of Combinatorics, 4 pp 229-235.
[15]	Stinson, D. R. (2004). Combinatorial Designs: Construction and Analysis, Springer_Verlag, New York, Inc., New York.

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John Chibayi, David Alila, Fredrick Onyango. (2017). Construction Procedure for Non-trivial T-designs. American Journal of Theoretical and Applied Statistics, 6(1), 52-60. https://doi.org/10.11648/j.ajtas.20170601.17

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ACS Style

John Chibayi; David Alila; Fredrick Onyango. Construction Procedure for Non-trivial T-designs. Am. J. Theor. Appl. Stat. 2017, 6(1), 52-60. doi: 10.11648/j.ajtas.20170601.17

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AMA Style

John Chibayi, David Alila, Fredrick Onyango. Construction Procedure for Non-trivial T-designs. Am J Theor Appl Stat. 2017;6(1):52-60. doi: 10.11648/j.ajtas.20170601.17

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@article{10.11648/j.ajtas.20170601.17,
  author = {John Chibayi and David Alila and Fredrick Onyango},
  title = {Construction Procedure for Non-trivial T-designs},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {1},
  pages = {52-60},
  doi = {10.11648/j.ajtas.20170601.17},
  url = {https://doi.org/10.11648/j.ajtas.20170601.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170601.17},
  abstract = {A t-design is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3…) occurs. The problem of finding all parameters (t, v, k, λt) for which t-(v, k, λt) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner t-designs are known for t ≥ 6 when v > k. The objective of this study therefore to develop new methods of constructing t-designs with t ≥ 3 and λ ≥1. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs.},
 year = {2017}
}

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TY  - JOUR
T1  - Construction Procedure for Non-trivial T-designs
AU  - John Chibayi
AU  - David Alila
AU  - Fredrick Onyango
Y1  - 2017/02/22
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170601.17
DO  - 10.11648/j.ajtas.20170601.17
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  - 52
EP  - 60
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170601.17
AB  - A t-design is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3…) occurs. The problem of finding all parameters (t, v, k, λt) for which t-(v, k, λt) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner t-designs are known for t ≥ 6 when v > k. The objective of this study therefore to develop new methods of constructing t-designs with t ≥ 3 and λ ≥1. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs.
VL  - 6
IS  - 1
ER  -

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Author Information

John Chibayi

Department of Statistics and Actuarial Science, MasenoUniverisity, Nairobi, Kenya
David Alila

Department of Mathematics, MasindeMuliro University of Science and Technology, Nairobi, Kenya
Fredrick Onyango

Department of Statistics and Actuarial Science, MasenoUniverisity, Nairobi, Kenya

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APA Style

John Chibayi, David Alila, Fredrick Onyango. (2017). Construction Procedure for Non-trivial T-designs. American Journal of Theoretical and Applied Statistics, 6(1), 52-60. https://doi.org/10.11648/j.ajtas.20170601.17

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ACS Style

John Chibayi; David Alila; Fredrick Onyango. Construction Procedure for Non-trivial T-designs. Am. J. Theor. Appl. Stat. 2017, 6(1), 52-60. doi: 10.11648/j.ajtas.20170601.17

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AMA Style

John Chibayi, David Alila, Fredrick Onyango. Construction Procedure for Non-trivial T-designs. Am J Theor Appl Stat. 2017;6(1):52-60. doi: 10.11648/j.ajtas.20170601.17

Copy | Download

@article{10.11648/j.ajtas.20170601.17,
  author = {John Chibayi and David Alila and Fredrick Onyango},
  title = {Construction Procedure for Non-trivial T-designs},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {1},
  pages = {52-60},
  doi = {10.11648/j.ajtas.20170601.17},
  url = {https://doi.org/10.11648/j.ajtas.20170601.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170601.17},
  abstract = {A t-design is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3…) occurs. The problem of finding all parameters (t, v, k, λt) for which t-(v, k, λt) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner t-designs are known for t ≥ 6 when v > k. The objective of this study therefore to develop new methods of constructing t-designs with t ≥ 3 and λ ≥1. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Construction Procedure for Non-trivial T-designs
AU  - John Chibayi
AU  - David Alila
AU  - Fredrick Onyango
Y1  - 2017/02/22
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170601.17
DO  - 10.11648/j.ajtas.20170601.17
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  - 52
EP  - 60
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170601.17
AB  - A t-design is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3…) occurs. The problem of finding all parameters (t, v, k, λt) for which t-(v, k, λt) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner t-designs are known for t ≥ 6 when v > k. The objective of this study therefore to develop new methods of constructing t-designs with t ≥ 3 and λ ≥1. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs.
VL  - 6
IS  - 1
ER  -

Copy | Download