Algorithm 1 Hi-DM encoding.

1:Data: Input bits $\mathbf{b}=(b_1,\dots ,b_k)$ 2:Result: The output amplitude sequence ${\mathbf{d}}^{\left(1\right)}$ 3:initialization ${\mathbf{d}}^{(L+1)}={\mathcal{D}}^{\left(L\right)}$ 4:for$\ell =L,\dots ,1$do 5: for $h=1,\dots ,T_{\ell }$ do 6:  Take ${\mathbf{b}}_t$ the first $k_{\ell }$ bits of $\mathbf{b}$; 7:  Remove ${\mathbf{b}}_t$ from $\mathbf{b}$; 8:  Let ${\mathcal{D}}_j^{(\ell )}$ be the DM in the component $d_h^{(\ell +1)}$ 9:  Use ${\mathcal{D}}_j^{(\ell )}$ to map ${\mathbf{b}}_t$ to a train of $N_{\ell }$ symbols and obtain the train of symbols $d_{(h-1)N_{\ell }+1}^{(\ell )},\dots ,d_{h{N}_{\ell }}^{(\ell )}$, where $d_i^{(\ell )}\in \left\{{\mathcal{D}}_1^{(\ell -1)},\dots ,{\mathcal{D}}_{M_{\ell }}^{(\ell -1)}\right\}$ if $\ell >1$ and $d_i^{(\ell )}\in \left\{1,3,\dots ,2M_1-1\right\}$ if $\ell =1$; 10:  Append the train of symbol ${\mathbf{d}}^{(\ell )}=({\mathbf{d}}^{(\ell )},d_{(h-1)N_{\ell }+1}^{(\ell )},\dots ,d_{h{N}_{\ell }}^{(\ell )})$; 11: end for 12:end for