# 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
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