A Separation of $\gamma$ and $b$ via Thue--Morse Words

We prove that for n ≥ 2, the size b(tn) of the smallest bidirectional scheme for the nth Thue–Morse word tn is n+ 2. Since Kutsukake et al. [SPIRE 2020] show that the size γ(tn) of the smallest string attractor for tn is 4 for n ≥ 4, this shows for the first time that there is a separation between the size of the smallest string attractor γ and the size of the smallest bidirectional scheme b, i.e., there exist string families such that γ = o(b).