# American Institute of Mathematical Sciences

May  2018, 12(2): 351-362. doi: 10.3934/amc.2018022

## Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4

 1 School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China 2 School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China

* Corresponding author: G. Chen(chenguangzhou0808@163.com)

Received  April 2017 Published  March 2018

Fund Project: The first author is supported by NSF grant No. 11501181, Science Foundation for Youths (Grant No. 2014QK05) and Ph.D.(Grant No. qd14140) of Henan Normal University.

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple $(v, \{3,4\}, λ)$-PBD for $λ = 7,9$ and $λ = 2k$, $k≥1$, are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given.

Citation: Guangzhou Chen, Yue Guo, Yong Zhang. Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4. Advances in Mathematics of Communications, 2018, 12 (2) : 351-362. doi: 10.3934/amc.2018022
##### References:
 [1] R. J. R. Abel and F. E. Bennett, Super-simple Steiner pentagon systems, Discrete Math., 156 (2008), 780-793.  doi: 10.1016/j.dam.2007.08.016. [2] R. J. R. Abel, F. E. Bennett and G. Ge, Super-Simple Holey Steiner pentagon systems and related designs, J. Combin. Designs, 16 (2008), 301-328.  doi: 10.1002/jcd.20171. [3] P. Adams, D. Bryant and A. Khodkar, On the existence of super-simple designs with block size 4, Aequationes Math., 51 (1996), 230-246.  doi: 10.1007/BF01833280. [4] T. L. Alderson and K. E. Mellinger, 2-dimensional optical orthogonal codes from singer groups, Discrete Appl. Math., 157 (2009), 3008-3019.  doi: 10.1016/j.dam.2009.06.002. [5] F. Amirzade and N. Soltankhah, Smallest defining sets of super-simple 2-(v, 4, 1) directed designs, Utilitas Mathematic, 96 (2015), 331-344. [6] I. Bluskov, New designs, J. Combin. Math. Combin. Comput., 23 (1997), 212-220. [7] I. Bluskov and H. Hämäläinen, New upper bounds on the minimum size of covering designs, J. Combin. Designs, 6 (1998), 21-41.  doi: 10.1002/(SICI)1520-6610(1998)6:1<21::AID-JCD2>3.0.CO;2-Y. [8] I. Bluskov and K. Heinrich, Super-simple designs with v ≤ 32, J. Statist. Plann. Inference, 95 (2001), 121-131.  doi: 10.1016/S0378-3758(00)00282-2. [9] H. Cao, K. Chen and R. Wei, Super-simple Balanced Incomplete block designs with block size 4 and index 5, Discrete Math., 309 (2009), 2808-2814.  doi: 10.1016/j.disc.2008.07.003. [10] H. Cao, F. Yan and R. Wei, Super-simple group divisible designs with blocks size 4 and index 2, J. Statist. Plann. Inference, 140 (2010), 2497-2503.  doi: 10.1016/j.jspi.2010.02.020. [11] G. Chen, K. Chen and Y. Zhang, Super-simple (5, 4)-GDDs of group type gu, Front. Math. China, 9 (2014), 1001-1018.  doi: 10.1007/s11464-014-0393-3. [12] G. Chen, Y. Zhang and K. Chen, Super-simple pairwise balanced designs with block sizes 3 and 4, Discrete Math., 340 (2017), 236-242.  doi: 10.1016/j.disc.2016.08.021. [13] K. Chen, On the existence of super-simple (v, 4, 3)-BIBDs, J. Combin. Math. Combin. Comput., 17 (1995), 149-159. [14] K. Chen, On the existence of super-simple (v, 4, 4)-BIBDs, J. Statist. Plann. Inference, 51 (1996), 339-350.  doi: 10.1016/0378-3758(95)00097-6. [15] K. Chen, Z. Cao and R. Wei, Super-simple balanced incomplete block designs with block size 4 and index 6, J. Statist. Plann. Inference, 133 (2005), 537-554.  doi: 10.1016/j.jspi.2004.01.013. [16] K. Chen, G. Chen, W. Li and R. Wei, Super-simple balanced incomplete block designs with block size 5 and index 3, Discrete Appl. Math., 161 (2013), 2396-2404.  doi: 10.1016/j.dam.2013.05.007. [17] K. Chen and R. Wei, Super-simple (v, 5, 4) designs, Discrete Appl. Math., 155 (2007), 904-913.  doi: 10.1016/j.dam.2006.09.009. [18] K. Chen and R. Wei, Super-simple (v, 5, 5) Designs, Des. Codes Crypt., 39 (2006), 173-187.  doi: 10.1007/s10623-005-3256-9. [19] K. Chen and R. Wei, On super-simple cyclic 2-designs, Ars Combin., 103 (2012), 257-277. [20] K. Chen and R. Wei, Super-simple cyclic designs with small values, J. Statist. Plann. Inference, 137 (2007), 2034-2044.  doi: 10.1016/j.jspi.2006.04.008. [21] K. Chen, Y. Sun and Y. Zhang, Super-simple balanced incomplete block designs with block size 4 and index 8, tilitas Mathematic, 91 (2013), 213-229. [22] F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.  doi: 10.1109/18.30982. [23] C. J. Colbourn and J. H. Dinitz (Editors), CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007. [24] M. Dehon, On the existence of 2-designs Sλ(2, 3, v) without repeated blocks, Discrete Math., 43 (1983), 155-171.  doi: 10.1016/0012-365X(83)90153-X. [25] H.-D. O. F. Gronau, D. L. Kreher and A. C. H. Ling, Super-simple (v, 5, 2) designs, Discrete Appl. Math., 138 (2004), 65-77.  doi: 10.1016/S0166-218X(03)00270-1. [26] H.-D. O. F. Gronau and R. C. Mullin, On super-simple 2-(v, 4, λ) designs, J. Combin. Math. Combin. Comput., 11 (1992), 113-121. [27] H.-D. O. F. Gronau, R. C. Mullin and Ch. Pietsch, The closure of all subsets of (3, 4, ..., 10) which include 3, Ars Combin., 41 (1995), 129-162. [28] S. Hartmann, On simple and super-simple transversal designs, J. Combin. Designs, 8 (2000), 311-320.  doi: 10.1002/1520-6610(2000)8:5<311::AID-JCD1>3.0.CO;2-1. [29] S. Hartmann, Superpure digraph designs, J. Combin. Designs, 10 (2000), 239-255.  doi: 10.1002/jcd.10013. [30] S. M. Johnson, A new upper bound for error-correcting codes, IEEE Trans. Inform. Theory, 8 (1962), 203-207. [31] A. Khodkar, Various super-simple designs with block size four, Australas. J. Combin., 9 (1994), 201-210. [32] H. K. Kim and V. Lebedev, Cover-free families, superimposed codes and key distribution patterns, J. Combin. Designs, 12 (2004), 79-91.  doi: 10.1002/jcd.10056. [33] A. C. H. Ling, X. J. Zhu, C. J. Colbourn and R. C. Mullin, Pairwise balanced designs with consecutive block sizes, Des. Codes Crypt., 10 (1997), 203-222.  doi: 10.1023/A:1008248521550. [34] H. Liu and L. Wang, Super-simple resolvable balanced incomplete block designs with block size 4 and index 4, Graphs Combin., 29 (2013), 1477-1488.  doi: 10.1007/s00373-012-1194-7. [35] D. R. Stinson, R. Wei and L. Zhu, New Constructions for perfect hash families and related structures using related combinatorial designs and codes, J. Combin. Designs, 8 (2000), 189-200.  doi: 10.1002/(SICI)1520-6610(2000)8:3<189::AID-JCD4>3.0.CO;2-A. [36] H. Wei, H. Zhang and G. Ge, Completely reducible super-simple designs with block size five and index two, Des. Codes Crypt., 76 (2015), 589-600.  doi: 10.1007/s10623-014-9979-8. [37] Y. Zhang, K. Chen and Y. Sun, Super-simple balanced incomplete block designs with block size 4 and index 9, J. Statist. Plann. Inference, 139 (2009), 3612-3624.  doi: 10.1016/j.jspi.2009.04.011.

show all references

##### References:
 [1] R. J. R. Abel and F. E. Bennett, Super-simple Steiner pentagon systems, Discrete Math., 156 (2008), 780-793.  doi: 10.1016/j.dam.2007.08.016. [2] R. J. R. Abel, F. E. Bennett and G. Ge, Super-Simple Holey Steiner pentagon systems and related designs, J. Combin. Designs, 16 (2008), 301-328.  doi: 10.1002/jcd.20171. [3] P. Adams, D. Bryant and A. Khodkar, On the existence of super-simple designs with block size 4, Aequationes Math., 51 (1996), 230-246.  doi: 10.1007/BF01833280. [4] T. L. Alderson and K. E. Mellinger, 2-dimensional optical orthogonal codes from singer groups, Discrete Appl. Math., 157 (2009), 3008-3019.  doi: 10.1016/j.dam.2009.06.002. [5] F. Amirzade and N. Soltankhah, Smallest defining sets of super-simple 2-(v, 4, 1) directed designs, Utilitas Mathematic, 96 (2015), 331-344. [6] I. Bluskov, New designs, J. Combin. Math. Combin. Comput., 23 (1997), 212-220. [7] I. Bluskov and H. Hämäläinen, New upper bounds on the minimum size of covering designs, J. Combin. Designs, 6 (1998), 21-41.  doi: 10.1002/(SICI)1520-6610(1998)6:1<21::AID-JCD2>3.0.CO;2-Y. [8] I. Bluskov and K. Heinrich, Super-simple designs with v ≤ 32, J. Statist. Plann. Inference, 95 (2001), 121-131.  doi: 10.1016/S0378-3758(00)00282-2. [9] H. Cao, K. Chen and R. Wei, Super-simple Balanced Incomplete block designs with block size 4 and index 5, Discrete Math., 309 (2009), 2808-2814.  doi: 10.1016/j.disc.2008.07.003. [10] H. Cao, F. Yan and R. Wei, Super-simple group divisible designs with blocks size 4 and index 2, J. Statist. Plann. Inference, 140 (2010), 2497-2503.  doi: 10.1016/j.jspi.2010.02.020. [11] G. Chen, K. Chen and Y. Zhang, Super-simple (5, 4)-GDDs of group type gu, Front. Math. China, 9 (2014), 1001-1018.  doi: 10.1007/s11464-014-0393-3. [12] G. Chen, Y. Zhang and K. Chen, Super-simple pairwise balanced designs with block sizes 3 and 4, Discrete Math., 340 (2017), 236-242.  doi: 10.1016/j.disc.2016.08.021. [13] K. Chen, On the existence of super-simple (v, 4, 3)-BIBDs, J. Combin. Math. Combin. Comput., 17 (1995), 149-159. [14] K. Chen, On the existence of super-simple (v, 4, 4)-BIBDs, J. Statist. Plann. Inference, 51 (1996), 339-350.  doi: 10.1016/0378-3758(95)00097-6. [15] K. Chen, Z. Cao and R. Wei, Super-simple balanced incomplete block designs with block size 4 and index 6, J. Statist. Plann. Inference, 133 (2005), 537-554.  doi: 10.1016/j.jspi.2004.01.013. [16] K. Chen, G. Chen, W. Li and R. Wei, Super-simple balanced incomplete block designs with block size 5 and index 3, Discrete Appl. Math., 161 (2013), 2396-2404.  doi: 10.1016/j.dam.2013.05.007. [17] K. Chen and R. Wei, Super-simple (v, 5, 4) designs, Discrete Appl. Math., 155 (2007), 904-913.  doi: 10.1016/j.dam.2006.09.009. [18] K. Chen and R. Wei, Super-simple (v, 5, 5) Designs, Des. Codes Crypt., 39 (2006), 173-187.  doi: 10.1007/s10623-005-3256-9. [19] K. Chen and R. Wei, On super-simple cyclic 2-designs, Ars Combin., 103 (2012), 257-277. [20] K. Chen and R. Wei, Super-simple cyclic designs with small values, J. Statist. Plann. Inference, 137 (2007), 2034-2044.  doi: 10.1016/j.jspi.2006.04.008. [21] K. Chen, Y. Sun and Y. Zhang, Super-simple balanced incomplete block designs with block size 4 and index 8, tilitas Mathematic, 91 (2013), 213-229. [22] F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.  doi: 10.1109/18.30982. [23] C. J. Colbourn and J. H. Dinitz (Editors), CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007. [24] M. Dehon, On the existence of 2-designs Sλ(2, 3, v) without repeated blocks, Discrete Math., 43 (1983), 155-171.  doi: 10.1016/0012-365X(83)90153-X. [25] H.-D. O. F. Gronau, D. L. Kreher and A. C. H. Ling, Super-simple (v, 5, 2) designs, Discrete Appl. Math., 138 (2004), 65-77.  doi: 10.1016/S0166-218X(03)00270-1. [26] H.-D. O. F. Gronau and R. C. Mullin, On super-simple 2-(v, 4, λ) designs, J. Combin. Math. Combin. Comput., 11 (1992), 113-121. [27] H.-D. O. F. Gronau, R. C. Mullin and Ch. Pietsch, The closure of all subsets of (3, 4, ..., 10) which include 3, Ars Combin., 41 (1995), 129-162. [28] S. Hartmann, On simple and super-simple transversal designs, J. Combin. Designs, 8 (2000), 311-320.  doi: 10.1002/1520-6610(2000)8:5<311::AID-JCD1>3.0.CO;2-1. [29] S. Hartmann, Superpure digraph designs, J. Combin. Designs, 10 (2000), 239-255.  doi: 10.1002/jcd.10013. [30] S. M. Johnson, A new upper bound for error-correcting codes, IEEE Trans. Inform. Theory, 8 (1962), 203-207. [31] A. Khodkar, Various super-simple designs with block size four, Australas. J. Combin., 9 (1994), 201-210. [32] H. K. Kim and V. Lebedev, Cover-free families, superimposed codes and key distribution patterns, J. Combin. Designs, 12 (2004), 79-91.  doi: 10.1002/jcd.10056. [33] A. C. H. Ling, X. J. Zhu, C. J. Colbourn and R. C. Mullin, Pairwise balanced designs with consecutive block sizes, Des. Codes Crypt., 10 (1997), 203-222.  doi: 10.1023/A:1008248521550. [34] H. Liu and L. Wang, Super-simple resolvable balanced incomplete block designs with block size 4 and index 4, Graphs Combin., 29 (2013), 1477-1488.  doi: 10.1007/s00373-012-1194-7. [35] D. R. Stinson, R. Wei and L. Zhu, New Constructions for perfect hash families and related structures using related combinatorial designs and codes, J. Combin. Designs, 8 (2000), 189-200.  doi: 10.1002/(SICI)1520-6610(2000)8:3<189::AID-JCD4>3.0.CO;2-A. [36] H. Wei, H. Zhang and G. Ge, Completely reducible super-simple designs with block size five and index two, Des. Codes Crypt., 76 (2015), 589-600.  doi: 10.1007/s10623-014-9979-8. [37] Y. Zhang, K. Chen and Y. Sun, Super-simple balanced incomplete block designs with block size 4 and index 9, J. Statist. Plann. Inference, 139 (2009), 3612-3624.  doi: 10.1016/j.jspi.2009.04.011.
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