Bootstrap percolation on a generalized Hamming cube \MakeUppercase{\romannumeral 2}

In this paper we investigate the critical probability $p_c(Q_n,r)$ for bootstrap percolation with the infection threshold $r$ on the $n$-dimensional hypercube $Q_n$ with vertex set $V(Q_n)=\{0,1{\}}^n$ and edges connecting the pairs at Hamming distance $1$. More precisely, by utilizing the techniques developed by Balogh, Bollob{á}s, and Morris (2009), we determine the first-order term of $p_c(Q_n,n^a)$ where $\frac{2}{3}<a<1$. Additionally, we obtain the critical probability $p_c(Q_{k,n},r)$ for bootstrap percolation with the infection threshold $r=\frac{N}{2}$ on the generalized $n$-dimensional hypercube $Q_{k,n}$ with vertex set $V(Q_{k,n})=\{0,1{\}}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots ,k$, where $k\ge 2$ and $N={\sum }_{i=1}^k\left(\genfrac{}{}{0px}{}{n}{i}\right)$. More precisely, we obtain the first-order term of $p_c(Q_{k,n},\frac{N}{2})$ and some bounds on the second-order term by extending the main theorem from Balogh, Bollob{á}s, and Morris (2009).