Construction of Some Resolvable <i>t</i>-designs

Alilah David

doi:doi:10.11648/j.sjams.20170501.17

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Construction of Some Resolvable t-designs

Alilah David

Published in Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 1)

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

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Abstract

The 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. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs. Additionally, an alternative approach for the construction of t-designs that provides a unified framework is also presented.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 5, Issue 1)
DOI	10.11648/j.sjams.20170501.17
Page(s)	49-53
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, Resolvable Designs, t-designs

References

[1]	Adhikari, B. (1967). On the symmetric differences of pairs of blocks of incomplete blockdesigns, Calcutta stat. Assoc. Bull 16, 45-48.
[2]	Blanchard, J. (1995a). A construction for Steiner 3-designs, Journal of combinatorialTheory A, 71, 60-67.
[3]	Blanchard, J. (1995c). A construction for orthogonal arrays with strength t ≥ 3, Discretemath 137, no. 1-3, 35-44.
[4]	Cameron, P., Maimani, H. R., Omidi, G. R., and Tayfeh-Rezaie, B. (2006). 3-designs PGL(2, q), Discrete Mathematics, 306, vol.23, 3063-3073.
[5]	Colbourn, C. (2002). Orthogonal arrays of strength three from regular 3-wise balanceddesigns.
[6]	Hartman, A. (1994). The fundamental construction for 3-designs. Discrete math 124, no. 1-3, 107-131.
[7]	Kageyama, S. (1991). A property of t-wise balanced designs, Ars. Combinatorial, 31, pp 237-238.
[8]	Mathon, R. and Rosa,A.(1985). Tables of paramenters of BIBds with r ≤ 41 includingexistence, enumeration and resolvability results. Annals of Discretemathematics 26, 275-308.
[9]	Mcsorley, J., and Soicher, L. (2005). Construction of t-designs from a t-wise balanceddesign, Eur. J. Comb.to appear.
[10]	Mohácsy, H., and Ray-Chaudhuri, D. (2001). A construction for infinite families ofSteiner 3-designs, Journal of Combinatorial Theory A, 94, 127-141.
[11]	Mohácsy, H., and Ray-Chaudhuri, D. (2002). Candelabra Systems and designs, Journal ofStatistical planning and Inference, 106, 419-448.
[12]	Mohácsy, H., and Ray-Chaudhuri, D. (2003). A construction for group divisible t-designswith strength t ≥ 2 and index unity, Journal of Statistical planning and Inference, 109, 167-177.
[13]	Onyango, O. (2010). Construction of t-(v, k, λt) designs, Journal of mathematicalscience, vol. 21 no. 4 pp 521-526.
[14]	Ray-Chaudhuri, D. and Wilson, R. (1975). “On t-designs”, Osaka, J. Math, 12, 737-744.
[15]	Stinson, D. (2004). Combinatorial Designs: Construction and Analysis, Springer_Verlag,New York, Inc., New York.
[16]	Wilson, R (1972a). An existence theory for pairwise balanced designs I. Compositiontheorems and Morphisms, Journal of Combinatorial Theory A, 13, 220-245.

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Alilah David. (2017). Construction of Some Resolvable t-designs. Science Journal of Applied Mathematics and Statistics, 5(1), 49-53. https://doi.org/10.11648/j.sjams.20170501.17

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ACS Style

Alilah David. Construction of Some Resolvable t-designs. Sci. J. Appl. Math. Stat. 2017, 5(1), 49-53. doi: 10.11648/j.sjams.20170501.17

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AMA Style

Alilah David. Construction of Some Resolvable t-designs. Sci J Appl Math Stat. 2017;5(1):49-53. doi: 10.11648/j.sjams.20170501.17

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@article{10.11648/j.sjams.20170501.17,
  author = {Alilah David},
  title = {Construction of Some Resolvable t-designs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {1},
  pages = {49-53},
  doi = {10.11648/j.sjams.20170501.17},
  url = {https://doi.org/10.11648/j.sjams.20170501.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170501.17},
  abstract = {The 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. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs. Additionally, an alternative approach for the construction of t-designs that provides a unified framework is also presented.},
 year = {2017}
}

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TY  - JOUR
T1  - Construction of Some Resolvable t-designs
AU  - Alilah David
Y1  - 2017/02/22
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170501.17
DO  - 10.11648/j.sjams.20170501.17
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 49
EP  - 53
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170501.17
AB  - The 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. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs. Additionally, an alternative approach for the construction of t-designs that provides a unified framework is also presented.
VL  - 5
IS  - 1
ER  -

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Author Information

Alilah David

Department of Mathematics, Masinde Muliro University of Science and Technology, Nairobi, Kenya

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APA Style

Alilah David. (2017). Construction of Some Resolvable t-designs. Science Journal of Applied Mathematics and Statistics, 5(1), 49-53. https://doi.org/10.11648/j.sjams.20170501.17

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ACS Style

Alilah David. Construction of Some Resolvable t-designs. Sci. J. Appl. Math. Stat. 2017, 5(1), 49-53. doi: 10.11648/j.sjams.20170501.17

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AMA Style

Alilah David. Construction of Some Resolvable t-designs. Sci J Appl Math Stat. 2017;5(1):49-53. doi: 10.11648/j.sjams.20170501.17

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@article{10.11648/j.sjams.20170501.17,
  author = {Alilah David},
  title = {Construction of Some Resolvable t-designs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {5},
  number = {1},
  pages = {49-53},
  doi = {10.11648/j.sjams.20170501.17},
  url = {https://doi.org/10.11648/j.sjams.20170501.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20170501.17},
  abstract = {The 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. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs. Additionally, an alternative approach for the construction of t-designs that provides a unified framework is also presented.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Construction of Some Resolvable t-designs
AU  - Alilah David
Y1  - 2017/02/22
PY  - 2017
N1  - https://doi.org/10.11648/j.sjams.20170501.17
DO  - 10.11648/j.sjams.20170501.17
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 49
EP  - 53
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20170501.17
AB  - The 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. In this study t-design is constructed by relating known BIB designs, combinatorial designs and algebraic structures with t-designs. Additionally, an alternative approach for the construction of t-designs that provides a unified framework is also presented.
VL  - 5
IS  - 1
ER  -

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