On symmetric BIBDs with the same 3-concurrence
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In a symmetric balanced incomplete block design (SBIBD), every pair of points appears \(\lambda \) times among all blocks. In this paper, we study the 3-concurrence of an SBIBD, i.e., the number of times that each triple of points appears. We try to find two distinct SBIBDs whose 3-concurrences are exactly the same. Existence of such a pair would give a non-trivial tight relative 3-design on two shells in the binary Hamming association scheme H(n, 2). We prove that such pairs of designs do not exist when \(\lambda =1,2\) or the block size is at least \((\lambda -1)(\lambda ^2-2)+2\). We also give criteria to check the existence of such pairs when the designs are given. For \(\lambda =3\), our computational results show the non-existence of such pairs with only two cases left unknown: (45,12,3) and (71,15,3).
KeywordsSymmetric BIBD 3-Concurrence Johnson graph
Mathematics Subject Classification05B05
We deeply thank Eiichi Bannai for suggesting the topic of this paper and for his advice and comments at various stages of this project. We thank Yaokun Wu for improving the presentation of the paper. We also thank Etsuko Bannai, Takuya Ikuta, Kyoungtark Kim and Yan Zhu for useful discussions. We are grateful to Ted Spence for the data of designs on his homepage, and especially for providing us the incidence matrices of some SBIBDs. This research was supported by NSFC Grant 11271257 and 11271255.
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