On symmetric BIBDs with the same 3-concurrence

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).