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Construction Procedure for Non-trivial T-designs

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.

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 1)
DOI 10.11648/j.ajtas.20170601.17
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), 2017. 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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  • 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

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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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    T1  - Construction Procedure for Non-trivial T-designs
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    AU  - David Alila
    AU  - Fredrick Onyango
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    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
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    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
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Author Information
  • Department of Statistics and Actuarial Science, MasenoUniverisity, Nairobi, Kenya

  • Department of Mathematics, MasindeMuliro University of Science and Technology, Nairobi, Kenya

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

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