Algorithm 1 Depth-first bit-labeling search.

Input:The assignment order I;

The label candidate set B.

Output: The 4 bit PGL schemes set $L^{*}$.

1: begin procedure

2: initialize: let $l^{{\mathcal{A}}^{*}}\left(x_{i_1}\right)=\left(0000\right)$, $B=B\setminus l^{{\mathcal{A}}^{*}}\left(x_{i_1}\right)$; set $t=2$;

3: find $S^t\subseteq B$;

4: if $S^t=Ø$ then

5:  go to 14;

6: end if

7: for $j=1;j<=|S^t|;j++$ do

8:  $l^{{\mathcal{A}}^{*}}\left(x_{i_t}\right)=s_j\in S^{\left(t\right)}$, $B=B\setminus l^{{\mathcal{A}}^{*}}\left(x_{i_t}\right)$, $t=t+1$;

9:  if $t=32$ then

10:   record the partial Gray labeling schemes ${\mathcal{L}}^{{\mathcal{A}}^{*}}:\{l^{{\mathcal{A}}^{*}}\left(x_{i_1}\right),l^{{\mathcal{A}}^{*}}\left(x_{i_2}\right),\dots ,l^{{\mathcal{A}}^{*}}\left(x_{i_{32}}\right)\}$ in $L^{*}$;

11:   go to 14;

12:  end if

13:  return to 3;

14:  $t=t-1$, $B=B\bigcup \left\{l^{{\mathcal{A}}^{*}}\left(x_{i_t}\right)\right\}$;

15: end for

16: return $L^{*}$.