Authors’ abstract: The existence of pairwise additive cyclic BIB designs with \(k=2\) and \(\lambda =1\) has been discussed in K. Matsubara and S. Kageyama [“The existence of two pairwise additive BIBD for any \(v\)”, J. Stat. Theory Pract. 7, No. 4, 783–790 (2013)]. However, the existence of \(2 \operatorname{PACB}(2^m,2,1)\) for any \(m\geq 5\) and \(m\equiv 1 \pmod 4\) has not been known in the literature. In this note, \(2 \operatorname{PACB}(32,2,1)\) is given by a computer and then the existence of \(2\operatorname{PACB}(2^{m}t,2,1)\) with any integer \(m\geq 2\) and any odd integer \(t\geq 1\) such that \(\gcd(t,27)\neq (3,9)\) can be proved.