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. Google Scholar

[2]

R. J. R. AbelF. 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. Google Scholar

[3]

P. AdamsD. 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. Google Scholar

[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. Google Scholar

[5]

F. Amirzade and N. Soltankhah, Smallest defining sets of super-simple 2-(v, 4, 1) directed designs, Utilitas Mathematic, 96 (2015), 331-344. Google Scholar

[6]

I. Bluskov, New designs, J. Combin. Math. Combin. Comput., 23 (1997), 212-220. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

H. CaoK. 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. Google Scholar

[10]

H. CaoF. 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. Google Scholar

[11]

G. ChenK. 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. Google Scholar

[12]

G. ChenY. 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. Google Scholar

[13]

K. Chen, On the existence of super-simple (v, 4, 3)-BIBDs, J. Combin. Math. Combin. Comput., 17 (1995), 149-159. Google Scholar

[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. Google Scholar

[15]

K. ChenZ. 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. Google Scholar

[16]

K. ChenG. ChenW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

K. Chen and R. Wei, On super-simple cyclic 2-designs, Ars Combin., 103 (2012), 257-277. Google Scholar

[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. Google Scholar

[21]

K. ChenY. Sun and Y. Zhang, Super-simple balanced incomplete block designs with block size 4 and index 8, tilitas Mathematic, 91 (2013), 213-229. Google Scholar

[22]

F. R. K. ChungJ. 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. Google Scholar

[23]

C. J. Colbourn and J. H. Dinitz (Editors), CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007. Google Scholar

[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. Google Scholar

[25]

H.-D. O. F. GronauD. 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. Google Scholar

[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. Google Scholar

[27]

H.-D. O. F. GronauR. C. Mullin and Ch. Pietsch, The closure of all subsets of (3, 4, ..., 10) which include 3, Ars Combin., 41 (1995), 129-162. Google Scholar

[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. Google Scholar

[29]

S. Hartmann, Superpure digraph designs, J. Combin. Designs, 10 (2000), 239-255. doi: 10.1002/jcd.10013. Google Scholar

[30]

S. M. Johnson, A new upper bound for error-correcting codes, IEEE Trans. Inform. Theory, 8 (1962), 203-207. Google Scholar

[31]

A. Khodkar, Various super-simple designs with block size four, Australas. J. Combin., 9 (1994), 201-210. Google Scholar

[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. Google Scholar

[33]

A. C. H. LingX. J. ZhuC. 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. Google Scholar

[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. Google Scholar

[35]

D. R. StinsonR. 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. Google Scholar

[36]

H. WeiH. 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. Google Scholar

[37]

Y. ZhangK. 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. Google Scholar

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. Google Scholar

[2]

R. J. R. AbelF. 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. Google Scholar

[3]

P. AdamsD. 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. Google Scholar

[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. Google Scholar

[5]

F. Amirzade and N. Soltankhah, Smallest defining sets of super-simple 2-(v, 4, 1) directed designs, Utilitas Mathematic, 96 (2015), 331-344. Google Scholar

[6]

I. Bluskov, New designs, J. Combin. Math. Combin. Comput., 23 (1997), 212-220. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

H. CaoK. 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. Google Scholar

[10]

H. CaoF. 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. Google Scholar

[11]

G. ChenK. 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. Google Scholar

[12]

G. ChenY. 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. Google Scholar

[13]

K. Chen, On the existence of super-simple (v, 4, 3)-BIBDs, J. Combin. Math. Combin. Comput., 17 (1995), 149-159. Google Scholar

[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. Google Scholar

[15]

K. ChenZ. 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. Google Scholar

[16]

K. ChenG. ChenW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

K. Chen and R. Wei, On super-simple cyclic 2-designs, Ars Combin., 103 (2012), 257-277. Google Scholar

[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. Google Scholar

[21]

K. ChenY. Sun and Y. Zhang, Super-simple balanced incomplete block designs with block size 4 and index 8, tilitas Mathematic, 91 (2013), 213-229. Google Scholar

[22]

F. R. K. ChungJ. 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. Google Scholar

[23]

C. J. Colbourn and J. H. Dinitz (Editors), CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007. Google Scholar

[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. Google Scholar

[25]

H.-D. O. F. GronauD. 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. Google Scholar

[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. Google Scholar

[27]

H.-D. O. F. GronauR. C. Mullin and Ch. Pietsch, The closure of all subsets of (3, 4, ..., 10) which include 3, Ars Combin., 41 (1995), 129-162. Google Scholar

[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. Google Scholar

[29]

S. Hartmann, Superpure digraph designs, J. Combin. Designs, 10 (2000), 239-255. doi: 10.1002/jcd.10013. Google Scholar

[30]

S. M. Johnson, A new upper bound for error-correcting codes, IEEE Trans. Inform. Theory, 8 (1962), 203-207. Google Scholar

[31]

A. Khodkar, Various super-simple designs with block size four, Australas. J. Combin., 9 (1994), 201-210. Google Scholar

[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. Google Scholar

[33]

A. C. H. LingX. J. ZhuC. 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. Google Scholar

[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. Google Scholar

[35]

D. R. StinsonR. 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. Google Scholar

[36]

H. WeiH. 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. Google Scholar

[37]

Y. ZhangK. 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. Google Scholar

[1]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036

[2]

Jamshid Moori, Amin Saeidi. Some designs and codes invariant under the Tits group. Advances in Mathematics of Communications, 2017, 11 (1) : 77-82. doi: 10.3934/amc.2017003

[3]

Michael Kiermaier, Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 2015, 9 (1) : 105-115. doi: 10.3934/amc.2015.9.105

[4]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161

[5]

Ivica Martinjak, Mario-Osvin Pavčević. Symmetric designs possessing tactical decompositions. Advances in Mathematics of Communications, 2011, 5 (2) : 199-208. doi: 10.3934/amc.2011.5.199

[6]

Peter Boyvalenkov, Maya Stoyanova. New nonexistence results for spherical designs. Advances in Mathematics of Communications, 2013, 7 (3) : 279-292. doi: 10.3934/amc.2013.7.279

[7]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[8]

Michael Braun, Michael Kiermaier, Reinhard Laue. New 2-designs over finite fields from derived and residual designs. Advances in Mathematics of Communications, 2019, 13 (1) : 165-170. doi: 10.3934/amc.2019010

[9]

Tran van Trung. Construction of 3-designs using $(1,\sigma)$-resolution. Advances in Mathematics of Communications, 2016, 10 (3) : 511-524. doi: 10.3934/amc.2016022

[10]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030

[11]

David Clark, Vladimir D. Tonchev. A new class of majority-logic decodable codes derived from polarity designs. Advances in Mathematics of Communications, 2013, 7 (2) : 175-186. doi: 10.3934/amc.2013.7.175

[12]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191

[13]

Karthikeyan Rajagopal, Serdar Cicek, Akif Akgul, Sajad Jafari, Anitha Karthikeyan. Chaotic cuttlesh: King of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019205

[14]

Qi Wang, Yue Zhou. Sets of zero-difference balanced functions and their applications. Advances in Mathematics of Communications, 2014, 8 (1) : 83-101. doi: 10.3934/amc.2014.8.83

[15]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53

[16]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451

[17]

Badal Joshi. A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1077-1105. doi: 10.3934/dcdsb.2015.20.1077

[18]

Yu Zhou. On the distribution of auto-correlation value of balanced Boolean functions. Advances in Mathematics of Communications, 2013, 7 (3) : 335-347. doi: 10.3934/amc.2013.7.335

[19]

Per Danzl, Ali Nabi, Jeff Moehlis. Charge-balanced spike timing control for phase models of spiking neurons. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1413-1435. doi: 10.3934/dcds.2010.28.1413

[20]

Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control & Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (54)
  • HTML views (252)
  • Cited by (0)

Other articles
by authors

[Back to Top]