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

Guangzhou Chen; Yue Guo; Yong Zhang

doi:10.3934/amc.2018022

Article Contents

2018, Volume 12, Issue 2: 351-362. Doi: 10.3934/amc.2018022

This issue Previous Article Completely regular codes by concatenating Hamming codes Next Article The weight distribution of quasi-quadratic residue codes

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

Received: April 2017

Published: May 2018

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 05B05, 51E05; Secondary: 62K10, 94C30.

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