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: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
F. Amirzade
and N. Soltankhah
, Smallest defining sets of super-simple 2-(v, 4, 1) directed designs, Utilitas Mathematic, 96 (2015)
, 331-344.
![]() ![]() |
|
I. Bluskov
, New designs, J. Combin. Math. Combin. Comput., 23 (1997)
, 212-220.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
K. Chen
, On the existence of super-simple (v, 4, 3)-BIBDs, J. Combin. Math. Combin. Comput., 17 (1995)
, 149-159.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
K. Chen
and R. Wei
, On super-simple cyclic 2-designs, Ars Combin., 103 (2012)
, 257-277.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
C. J. Colbourn and J. H. Dinitz (Editors), CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC, Boca Raton, FL, 2007.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
S. Hartmann
, Superpure digraph designs, J. Combin. Designs, 10 (2000)
, 239-255.
doi: 10.1002/jcd.10013.![]() ![]() ![]() |
|
S. M. Johnson
, A new upper bound for error-correcting codes, IEEE Trans. Inform. Theory, 8 (1962)
, 203-207.
![]() ![]() |
|
A. Khodkar
, Various super-simple designs with block size four, Australas. J. Combin., 9 (1994)
, 201-210.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |