A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized toα-resolvability and affine α-resolvability in 1963. When α = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and α-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine α-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L2), cyclic. First, it could be proved that an affine α-resolvable cyclic 2-associate PBIB design does not exist for any α ≥ 1. Also, Kageyama proved the non-existence of an affine α-resolvable triangular design for 1 ≤ α ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L2 designs with parameters υ ≤ 100 and r, k ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L2 design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L2 design which is left as only one unknown among L2 designs.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 2) |
| DOI | 10.11648/j.ijtam.20200602.12 |
| Page(s) | 28-30 |
| 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), 2020. Published by Science Publishing Group |
Affine Resolvability, Resolvability, PB Design, L2 Design
| [1] | T. Beth, D. Jungnickel and H. Lenz, Design Theory, Volume 1, 2nd Edition, Cambridge Univ. Press, UK, 1999. |
| [2] | R. C. Bose, A note on the resolvability of balanced incomplete block designs, Sankhy¯ a A 6 (1942), 105-110. |
| [3] | T. Cali´ nski and S. Kageyama, Block Designs: A Randomization Approach, Vol. II: Design, Lecture Notes in Statistics 170, Springer, New York, 2003. |
| [4] | W. H. Clatworthy, Tables of Two-Associate-Class Partially Balanced Designs, NBS Applied Mathematics Series 63, U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., 1973. |
| [5] | P. J. Dukes and E. R. Lamken, Constructions and uses of incomplete pairwise balanced designs. Des. Codes Cryptogr. 12 (2019), 2729-2751. |
| [6] | S. Furino, Y. Miao and J. Yin, Frames and Resolvable Designs, CRC Press, Boca Raton FL, 1996. |
| [7] | S. Kadowaki and S. Kageyama, A 2-resolvable BIBD(10, 15, 6, 4, 2) does not exist, Bulletin of the ICA. 53 (2008), 87-98. |
| [8] | S. Kadowaki and S. Kageyama, Existence of affine α-resolvable PBIB designs with some constructions, Hiroshima Math. J. 39 (2009), 293-326. Erratum, Hiroshima Math. J. 40 (2010), p. 271. |
| [9] | E. Kramer and D. Kreher, t-Wise balanced designs. In: Handbook of Combinatorial Designs (ed. by C. J. Colbourn and J. H. Dinitz), Second edition, CRC Press, pp. 657-663. |
| [10] | O. P. Popoola and B. A. Oyejola, Construction of congruent classes of pairwise balanced designs using lotto designs. Annals. Computer Science Series. 1 (2019), 127-134. |
| [11] | D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Dover, New York, 1988. |
| [12] | S. S. Shrikhande and D. Raghavarao, Affine α-resolvable incomplete block designs, Contributions to Statistics, Volume presented to Professor P. C. Mahalanobis on his 75th birthday, Pergamon Press, Oxford and Statistical Publishing Society, Calcutta, 1963, 471-480. |
APA Style
Satoru Kadowaki, Sanpei Kageyama. (2020). A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa. International Journal of Theoretical and Applied Mathematics, 6(2), 28-30. https://doi.org/10.11648/j.ijtam.20200602.12
ACS Style
Satoru Kadowaki; Sanpei Kageyama. A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa. Int. J. Theor. Appl. Math. 2020, 6(2), 28-30. doi: 10.11648/j.ijtam.20200602.12
AMA Style
Satoru Kadowaki, Sanpei Kageyama. A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa. Int J Theor Appl Math. 2020;6(2):28-30. doi: 10.11648/j.ijtam.20200602.12
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Download@article{10.11648/j.ijtam.20200602.12,
author = {Satoru Kadowaki and Sanpei Kageyama},
title = {A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {6},
number = {2},
pages = {28-30},
doi = {10.11648/j.ijtam.20200602.12},
url = {https://doi.org/10.11648/j.ijtam.20200602.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20200602.12},
abstract = {A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized toα-resolvability and affine α-resolvability in 1963. When α = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and α-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine α-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L2), cyclic. First, it could be proved that an affine α-resolvable cyclic 2-associate PBIB design does not exist for any α ≥ 1. Also, Kageyama proved the non-existence of an affine α-resolvable triangular design for 1 ≤ α ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L2 designs with parameters υ ≤ 100 and r, k ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L2 design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L2 design which is left as only one unknown among L2 designs.},
year = {2020}
}
TY - JOUR T1 - A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa AU - Satoru Kadowaki AU - Sanpei Kageyama Y1 - 2020/06/30 PY - 2020 N1 - https://doi.org/10.11648/j.ijtam.20200602.12 DO - 10.11648/j.ijtam.20200602.12 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 28 EP - 30 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20200602.12 AB - A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized toα-resolvability and affine α-resolvability in 1963. When α = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and α-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine α-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L2), cyclic. First, it could be proved that an affine α-resolvable cyclic 2-associate PBIB design does not exist for any α ≥ 1. Also, Kageyama proved the non-existence of an affine α-resolvable triangular design for 1 ≤ α ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L2 designs with parameters υ ≤ 100 and r, k ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L2 design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L2 design which is left as only one unknown among L2 designs. VL - 6 IS - 2 ER -
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